Mr. Trump, in a rambling, muddled answer on a conservative podcast, was criticizing President Biden’s leadership when he abruptly brought up his skepticism over the administration’s continued military aid to Ukraine.

“I think Zelensky is one of the greatest salesmen I’ve ever seen,” Mr. Trump said, repeating a statement he has made frequently. “Every time he comes in, we give him $100 billion. Who else got that kind of money in history? There’s never been. And that doesn’t mean I don’t want to help him, because I feel very badly for those people. But he should never have let that war start. That war is a loser.”

Mr. Trump has suggested before that blame for the widespread destruction caused by the Russian invasion rests with the Ukrainian president. Mr. Zelensky, he has said, should have cut a deal with Mr. Putin to avoid the invasion.

“Those cities are gone, they’re gone, and we continue to give billions of dollars to a man who refused to make a deal,” Mr. Trump said at a rally last month in Mint Hill, N.C.

On the podcast, after lamenting the devastation in Ukraine, Mr. Trump pointed the finger at Mr. Biden. Speaking with the podcast’s host, Patrick Bet-David, a conservative finance entrepreneur, Mr. Trump denigrated Mr. Biden’s intelligence by claiming that with a smarter president, the war “would have been easy to settle.”

Asked about the comments, the Trump campaign declined to clarify.

Mr. Trump’s remarks pointed to the continued concern by Democrats and American allies over how Mr. Trump might approach the conflict if he is elected in November. When Mr. Trump, in his debate with Vice President Kamala Harris last month, was pressed on the question of whether he wanted Ukraine to win the war, he did not say.

Mr. Trump has expressed admiration for Mr. Putin and speaks often about their closeness. This week, he twice dodged questions about a report that he had spoken to Mr. Putin after his presidency had ended but said that it would have been a “smart thing” if he had done so.

Though their ties have been under scrutiny since U.S. intelligence agencies concluded that Mr. Putin ordered an election interference effort to help Mr. Trump in his 2016 campaign, Mr. Trump’s frequent criticism of military and financial aid to Ukraine has raised new concerns.

Mr. Trump often insists that his relationship with Mr. Putin will allow him to end the war in Ukraine swiftly. But he has not addressed whether he believes that Ukraine will have to cede territory to Russia to do so.

Mr. Trump and Mr. Zelensky met last month at Trump Tower in Manhattan, their first meeting in five years. There, Mr. Trump told reporters that he wanted to end the war with “a fair deal for both sides.”

When the two made a joint appearance on Fox News, Mr. Zelensky made it clear that it was Mr. Putin who had started the war.

“This war shouldn’t be started, and I think that the problem that Putin killed so many people and, of course, we need to do everything to pressure him to stop this war,” Mr. Zelensky said. “He is on our territory, that’s the most important to understand. He is on our territory, and how to stop the war to pressure him as we can. As we can — we have to do it.”

“The man calls himself the father of I.V.F. I mean, what does that even mean?” Ms. Harris said after playing a video montage of Mr. Trump bragging about appointing Supreme Court justices who voted to overturn the Roe v. Wade decision that legalized abortion nationwide. “He is the one who, by the way, is responsible for it being at risk in the first place.”

The vice president added: “When you listen to Donald Trump talk, it becomes increasingly clear, I think, he has no idea what he’s talking about.”

Ms. Harris’s stinging remarks about abortion rights and in vitro fertilization came in Ashwaubenon, a Green Bay suburb in the heart of northeast Wisconsin, a region crucial to both candidates’ prospects in the state.

Earlier, on the other side of the state in La Crosse, Ms. Harris said her Republican opponent was “gaslighting” Americans with his effort to rewrite the history of Jan. 6 as a mostly peaceful day. He has falsely argued that the violence was instigated not by his supporters but by forces opposed to him.

“We here know Jan. 6 was a tragic day, it was a day of terrible violence,” Ms. Harris told a crowd organizers estimated at about 3,000 at the University of Wisconsin-La Crosse campus gymnasium. “He called it, quote, a day of love. But it points out something that everyone here knows. The American people are exhausted with his gaslighting. Exhausted with his gaslighting. Enough. We are ready to turn the page.”

Mr. Trump had been pressed on by a voter during a town-hall event broadcast Wednesday on Univision to explain why he should earn back the trust of Americans who grew disillusioned with him because of his actions, including on Jan. 6.

“That was a day of love from the standpoint of the millions,” Mr. Trump said. “It could have been the largest group I’ve ever spoken before. They asked me to speak and I went and I spoke.”

The mob attack on Jan. 6 in fact injured roughly 150 law enforcement officers and led to the deaths of several others.

Ms. Harris was joined earlier in Milwaukee and then in La Crosse by Mark Cuban, the celebrity tech billionaire who has owned a professional basketball team and hosted a reality television show.

Mr. Cuban, who appeared to be reading his remarks from his phone in La Crosse, said Mr. Trump was speaking “gibberish” about tariffs and suggested the former president was confused about his own proposals.

“Back in the ’90s and early 2000s, he was a little bit coherent when he talked about trade policies,” Mr. Cuban told the crowd. “The way he talks about trade policy, something is a little bit lost.”

Mr. Cuban, who last year endorsed Nikki Haley in the Republican presidential primary race before backing Ms. Harris, subsequently explained that tariffs would increase prices on an array of goods. He then described Mr. Trump as “the Grinch that wants to steal your Christmas.”

Ms. Harris’s trip was infused with efforts to ingratiate herself with Wisconsinites in ways large and small. In La Crosse, signage behind the stage featured an outline of a block of cheese. The Ashwaubenon rally took place in a convention center next to Lambeau Field, home to the locally beloved Green Bay Packers.

One of the warm-up speakers there, Kristin Lyerly, a Democrat running for an open congressional seat in northeast Wisconsin, led the crowd in an enthusiastic cheer of “Go Pack Go,” a staple at Packers games and many other social events in the state.

It was Ms. Harris’s sixth trip to the state since she replaced President Biden as the Democratic presidential nominee this summer. The three stops on Thursday represented an increase in her campaign’s typical daily velocity, and signaled an urgency to reach as many voters as possible in the final weeks before Election Day.

Polling in Wisconsin, as in other battleground states, shows a neck-and-neck race. Ms. Harris had an advantage of four percentage points in the most recent poll from Marquette University Law School, one of the most reputable surveys of the state, though most other polls have shown her and Mr. Trump within one or two points of each other.

Before Ms. Harris began her first planned event of the day at the University of Wisconsin-Milwaukee, news broke that Israel had killed Yahya Sinwar, the Hamas leader seen as the architect of last year’s Oct. 7 attack that killed roughly 1,200 people in Israel.

Ms. Harris broke from her planned schedule and delivered brief remarks, noting Mr. Sinwar’s death and calling for an end to the war in Gaza.

Jay Senter contributed reporting from La Crosse, Wis.

The New York Times has analyzed changes in the organization’s candidate ratings in every general election since 2010. This year’s analysis began with data provided by the nonprofit Everytown for Gun Safety, which supports tighter restrictions on guns. The Times independently verified the data and ran its own calculations.

The decline in the number of top ratings happened entirely among Republican candidates, 63 percent of whom received A’s, compared with 71 percent in 2022. No Democrat received an A grade in 2022, though, in a twist, one did this year: Representative Mary Peltola of Alaska.

The share of Republicans who received question-mark ratings — indicating no clear public record on gun rights, and no response to the N.R.A.’s questionnaire — jumped to 31 percent from 24 percent in 2022.

Justin Wagner, Everytown’s senior director of investigations, who ran the group’s analysis, said the number of question-mark ratings suggested that fewer candidates viewed the N.R.A. as an interest group whose support was worth courting, even if their policy views might align.

Only about 2 percent of Republican candidates — nine people — received D’s or F’s, indicating that they actively supported major gun-control measures.

Among Democrats, about 87 percent — up from 81 percent in 2022 — received F ratings, meaning the N.R.A. considers them a “true enemy of gun owners’ rights.”

F-rated candidates support measures that the N.R.A. opposes — for example, a ban on assault weapons, or an expansion of red-flag laws or background checks — but the grade doesn’t mean they want to ban most guns. Indeed, Vice President Kamala Harris, who supports all three of those measures and whom the N.R.A. is campaigning against, has made a point of emphasizing that she owns a handgun for self-defense.

Just over 10 percent of Democrats received a question-mark rating, suggesting that Democrats have been more consistently vocal about supporting gun control than Republicans have been about opposing it.

Among Democrats, B’s and C’s accounted for less than 1 percent, and 2.4 percent of candidates had D’s. Among Republicans, 3.3 percent had B’s, while C’s and D’s each accounted for less than 1 percent.

As is the case every election cycle, the shifts were driven mainly by new candidates, while most incumbents received the same grade as they had the last time they ran. Only five incumbents — four Republicans and one Democrat — received higher grades this year than they previously had, while four — three Democrats and one independent, Senator Angus King of Maine, who caucuses with Democrats — received lower ones.

Randy Kozuch, the chairman of the N.R.A. Political Victory Fund — the group’s political action committee — said on Thursday: “We are proud to stand with candidates up and down the ballot who support the Second Amendment. This November, we are committed to electing Donald Trump as president and electing pro-gun majorities in the U.S. House and Senate.”

Mr. Kozuch did not respond to the specific findings of the ratings analysis.

Now, some Trump advisers and allies say privately they are concerned that the dynamic may be repeating itself four years later. They worry that Mr. Trump’s impetuousness and scattershot style on the campaign trail needlessly risk victory in battleground states where the margin for error is increasingly narrow.

At a time when his opponent, Vice President Kamala Harris, has stepped up her attacks on him as “unstable,” Mr. Trump has struggled to publicly hone his message by veering off script and ramping up personal attacks on Ms. Harris that allies have urged him to rein in.

“When he’s good, he’s great, and when he’s off message, he’s not so great,” said David Urban, a Trump adviser. “I don’t think anyone is really changing their mind at this point, but when he distracts from his biggest, broadest messaging, it’s counterproductive because the Harris campaign uses it to turn out their voters.”

During a speech on Saturday in California, he described mail-in ballots as “so corrupt,” reviving one of his false attacks on the 2020 election results, and did a play-by-play of his internal thoughts when he watched SpaceX, Elon Musk’s spaceflight company, fly a rocket back onto its launch site.

On Sunday, in response to a question on Fox News about the possibility of foreign adversaries’ meddling in the election, he reverted to autocratic language by saying “the bigger problem is the enemy from within.” On Monday, he halted a town-hall event in suburban Philadelphia after five questions when two people in the crowd needed medical attention. He spent roughly the next half-hour playing D.J., swaying and grooving in front of his crowd to a playlist he curated from the stage. “Let’s just listen to music,” he said.

Last week, he canceled a CBS interview on “60 Minutes,” in which he and Ms. Harris were both scheduled to appear — and has not stopped talking about it. He complained about it during events in Detroit and Reno, Nev., and again on Monday in a social media post at 1:12 a.m.

At the Economic Club of Chicago on Tuesday, he answered a question about whether he would break up Google by complaining about a Justice Department lawsuit against Virginia election officials. When he was reminded the question was about Google, he said he “called the head of Google the other day” to grouse about the difficulty of finding positive news stories about his campaign on the company’s search page.

During the same event, Mr. Trump suggested his digressions were part of a communication strategy when his interviewer tried to focus on a question.

“You’ve got to be able to finish a thought because it’s very important,” Mr. Trump told the interviewer, John Micklethwait, the editor in chief of Bloomberg News. “This is big stuff we’re talking about. You can’t go that quickly.”

Mr. Micklethwait pointed out that Mr. Trump had started talking about reserve currency and then moved to a story about President Emmanuel Macron of France, among other digressions. “It’s called ‘the weave,’” Mr. Trump said, using a phrase he often uses to describe his speaking style, as he waved his hand in front of him to suggest he was connecting various dots.

Ms. Harris and her campaign have gone on the offensive by using Mr. Trump’s rambling against him, attacking him in ads, in speeches on the campaign trail and in interviews.

Internal Harris campaign research showed that one of the most effective ways to persuade voters to support the vice president was by portraying Mr. Trump as unstable and Ms. Harris as a steady leader who would strengthen America’s security, according to two Harris officials who insisted on anonymity to describe private data.

In the past two weeks, the Harris campaign has flooded the airwaves in battleground states with a pair of television ads to underscore these themes. One spot features warnings from Mr. Trump’s former top defense officials to paint him as “too big of a risk.” Another features endorsements for Ms. Harris from a bipartisan group of national security officials.

“Even former Trump administration officials agree there’s only one candidate fit to lead our nation — and that’s Kamala Harris,” the narrator says.

The Harris campaign criticized Mr. Trump’s appearance at the Economic Club of Chicago, saying he displayed “unstable behavior” and was “angry and unfocused as he rambled on and on.”

“No one has ever been more dangerous to this country than Donald Trump,” Gov. Tim Walz of Minnesota, the Democratic vice-presidential nominee, said on Monday at a campaign event in Wisconsin.

Ms. Harris said the former president was “quite unstable and unfit” during an interview on Monday with The Shade Room, a digital entertainment publication. In a second interview that day with the independent Black journalist Roland Martin, Ms. Harris pointed to Mr. Trump’s false claims that Haitian migrants were eating their neighbors’ pets.

“This man is dangerous,” she said.

Anna Kelly, a spokeswoman for the Republican National Committee, which is supporting the Trump campaign, said Mr. Trump’s “message is clear and consistent: President Trump’s agenda for America’s working men and women will fix our broken economy to lower costs and secure the border to make our communities safe.”

In Prescott Valley, Ariz., on Sunday, Mr. Trump’s scripted remarks hewed tightly to the anti-immigration message that has become central to his campaign. He stayed on track for the first half-hour of the event before taking a more scenic route to the finish.

After about 25 minutes, he told the crowd he wanted to tell “one quick story” about a friend with a car plant in Mexico.

But he never finished his tale. Instead, he lost the thread one minute later as he complained that if he mispronounced one word he would be accused of being “cognitively impaired.” Then, he botched the phrase by saying President Biden was the one who was “cognitively repaired” and referred to the election as three and a half months away, not three and a half weeks.

About 20 minutes later, Mr. Trump seemed ready to wrap up his speech. He promised the crowd would see him again soon and said he was thinking about residents on the East Coast suffering after the recent storms.

“So in closing,” Mr. Trump continued, “I just want to say Kamala Harris is a radical left Marxist rated even worse than Bernie Sanders or Pocahontas.”

He proceeded to speak for 17 more minutes.

Taylor Robinson contributed reporting.

“This moment gives us an opportunity to finally end the war in Gaza, and it must end such that Israel is secure, the hostages are released, the suffering in Gaza ends, and the Palestinian people can realize their right to dignity, security, freedom and self-determination,” Ms. Harris said at the University of Wisconsin-Milwaukee during a campaign visit. “It is time for the day after to begin without Hamas in power.”

Ms. Harris, who spoke for three minutes, did not answer questions from reporters asking if she would directly call on Israel to end the war in Gaza. She gave her remarks just minutes after President Biden — who was aboard Air Force One en route to a diplomatic visit in Berlin focused on ending the conflict in Ukraine — issued a public statement on the killing. Ms. Harris and her advisers were told to wait until Mr. Biden’s statement went public before stepping in front of the cameras, according to a senior administration official briefed on the plans.

Still, the appearance of the vice president stepping out in front of the cameras before the president underscored just how much scrutiny has been directed at her support of Israel’s military campaign in Gaza. The war has sharply divided Americans, especially younger ones, and continues to threaten her standing with key voters in battleground states.

As a presidential candidate, Ms. Harris has defied political pressure to break from her administration’s support of Israel and Mr. Biden’s increasingly strained embrace of Prime Minister Benjamin Netanyahu.

In interviews and statements, she has offered small departures from Mr. Biden but has been careful not to hint that a Harris presidency would drastically change the nature of the United States’ relationship with Israel.

In an interview last week with the CBS news program “60 Minutes,” Ms. Harris avoided repeated questions about how to end the yearlong war in Gaza, or whether the administration had lost all sway with Mr. Netanyahu. Also last week, she was on the secure line when Mr. Biden and Mr. Netanyahu spoke by phone for the first time in months, as U.S. national security officials grew more worried about a widening war in the Middle East.

Khaled Elgindy, a senior fellow at the Middle East Institute, said Ms. Harris’s strategy in the weeks ahead should not be limited to talking points but should demonstrate what she would do as commander in chief to bring an end to the conflict.

“It’s only going to help them if they can bring it to a close, and not just in aspiration terms but actually take steps that align with what they say,” Mr. Elgindy said. “This is a great opportunity to make distance between the failure of the last year of the Biden White House and what she would do differently.”

There is no sign of that yet from Ms. Harris, who on Thursday emphasized that the war had caused “unconscionable suffering of many innocent Palestinians and greater instability throughout the Middle East,” but largely still followed Mr. Biden’s lead.

In his statement, Mr. Biden said that Mr. Sinwar’s killing had rendered Hamas “no longer capable of carrying out another Oct. 7.” The president said that he would be speaking soon with Mr. Netanyahu to discuss the return of hostages and “ending this war once and for all” — a more direct tack than Ms. Harris took in her remarks.

Jake Sullivan, Mr. Biden’s national security adviser, told reporters on Air Force One that Mr. Sinwar’s death should “present an opportunity to find a way forward that gets the hostages home, brings the war to an end, brings us to a ‘day after.’” Mr. Sullivan said the administration was planning to talk about the future of Gaza with its Israeli counterparts.

Before she spoke in Milwaukee, Ms. Harris’s motorcade was greeted by protesters, who yelled “Free Palestine” and “We charge you with genocide” as her S.U.V. drove past.

There is growing fear among Democrats that Ms. Harris’s decision not to take a harder line against Israel will cost her the votes she needs to win the election, particularly in a battleground state like Michigan, home to a significant population of Arab American and Muslim voters.

Leaders of the Uncommitted National Movement, the group that mobilized hundreds of thousands of primary voters to cast protest ballots against Mr. Biden and said last month that it would not endorse Ms. Harris, signaled that a cease-fire would not be enough to regain its confidence.

“If we keep sending bombs to the Israeli government, Netanyahu will continue to collectively punish Palestinian civilians as he has for years,” Layla Elabed, the group’s co-founder, said in a statement. “Ending the war means ending our role in funding Israel’s militarism.”

Representative Haley Stevens of Michigan said Democrats wanted to hear Ms. Harris continue to position herself as a “responsible commander in chief” who would keep American troops out of a conflict in the Middle East while supporting both Israel and the right of Palestinians to determine their own future.

But Ms. Stevens cautioned that she did not believe Ms. Harris could win back the votes of voters who “don’t want Israel to exist.”

She added: “You have to understand the pushing here is really for a one-state solution on behalf of some. And that is never anything that the Democratic Party nor our Democratic nominee is going to articulate.”

Erica Green, Zolan Kanno-Youngs and Nicholas Nehamas contributed reporting from Washington.

On the Screen

The ad begins with an image of Mr. Trump, as president, seen from behind while exiting a doorway at the White House as Marine guards step aside for him.

A man with a deep baritone gives voice to yellow-green headlines that flash through various forms of taxation that Mr. Trump says he wants to get rid of: “One man believes in no tax on tips. No tax on Social Security. No tax on overtime.”

As he speaks, the ad cuts through a quick succession of images of working-class and older Americans: a waitress carrying a tray of dirty dishes; an older couple at the kitchen table; a snapshot of Mr. Trump seated at his Oval Office desk flanked by Irving Locker, a D-Day veteran, and his wife, Bernice; a woman putting on an apron in a restaurant kitchen, a welder, a delivery person on a bike, a hospital worker in a surgical mask and scrubs, a man in a Dallas Cowboys shirt operating a machine.

Along the way, a clip shows Mr. Trump in a hard hat, flashing a thumbs-up. The words “American workers have been forgotten” soon follow, overlaid on a clip of Vice President Kamala Harris and her sister, Maya, laughing uproariously. That headline lingers over an image of a taxi driver behind the wheel, and then a truck driver filling his tank and looking up at the sun.

The ad cuts to video of Mr. Trump in the moments after he was nearly assassinated in July in Pennsylvania, pumping his fist and mouthing the word “fight,” as the headline “One man will refuse to fall” appears.

A billowing American flag briefly fills the screen, and then the headline is replaced with a last one — “For American workers to continue to stand” — over a final montage: Trump supporters, a retail clerk, a waitress, a man in a hard hat, Mr. Trump embracing victims of a tornado in Alabama, and a taxi passenger paying her smiling driver.

The Script

Narrator:

“One man believes in no tax on tips, no tax on Social Security, and now, no tax on overtime.”

TRump

“The people who work overtime are among the hardest-working citizens in our country. And for too long, no one in Washington has been looking out for them.”

Narrator

“And one man will refuse to fall.”

CROWD

“U.S.A.! U.S.A.!”

NARRATOR

“So America and its workers can continue to stand great again.”

Accuracy

Mr. Trump has indeed called for a series of tax cuts during his 2024 presidential run, including on income from tips, overtime and Social Security.

The Takeaway

Mr. Trump has been trying to peel off more working-class voters, especially Black and Latino Americans who have historically voted overwhelmingly for Democrats. One of the ways he has tried to appeal to them is by promising big tax breaks. Working-class voters would benefit most from exempting tips and overtime pay from income tax. One sign that the proposals could be making inroads: Ms. Harris copied the no-tax-on-tips proposal in August during a campaign stop of her own in Nevada.

The ad ends by trying to harness the emotional resonance of Mr. Trump’s striking response after the assassination attempt, portraying his fighting spirit as something he will harness for the benefit of American workers.

The ad was paid for by a super PAC, Right for America, whose biggest donors are the billionaires Ike and Laura Perlmutter, who have given it $25 million.

They are part of a growing set of affluent, mostly left-leaning New Yorkers taking advantage of an unusual quirk in state law that allows second-home owners to vote from their country cottages, vacation homes and Hamptons houses that just happen to dot some of the most competitive congressional districts in the country.

Call it the rise of weekender politics.

It is no accident. With a half-dozen competitive districts, New York has taken center stage in the fight for Congress, and Democratic organizers believe that registering a fraction of the tens of thousands of New Yorkers who own second homes could help tip a Republican majority to a Democratic one.

As of late September, they had helped nearly 2,500 voters shift their registration from New York City into one of the state’s swing districts, according to data provided by MoveIndigo, a group spearheading the effort. The numbers are expected to grow as voting nears.

The sprawling 19th District has seen the biggest shift, with 1,040 voters newly registered at second homes in the Hudson Valley and Catskills regions, according to the group. Representative Marc Molinaro, a Republican, won the seat by 4,500 votes in 2022, and is running neck and neck with Josh Riley, a Democrat.

“If you look at the map of our state, it’s like tomato soup with a couple blueberries floating in it,” said Ms. Cramer, who already cast her vote for Mr. Riley in Sullivan County, where she owns a second home and enjoys skiing and hiking.

“I realized, oh, my vote could make a difference,” she added.

Voter registration pushes are nothing new. Though they have not targeted second-home owners, Republicans in Hudson Valley are encouraging sign-ups in growing Orthodox Jewish enclaves, and Democrats have targeted college campuses. A smattering of second-home owners have taken it upon themselves to register outside the city in recent years, especially during the Covid pandemic.

Still, it is hard to find a precedent for a campaign quite like this one.

MoveIndigo initially set out with an even more ambitious project: to try to persuade Democrats who have been thinking about moving to settle in one of dozens of swing congressional districts across the country. The group’s website offers personalized recommendations and gauzy pages advertising the amenities of settling in, say, southern Nevada, which boasts both “breathtaking natural landscapes” and a slew of competitive races.

But after the 2022 midterms, when unexpected Democratic losses in New York and California handed Republicans a narrow House majority, its founders saw a unique confluence of factors making it possible to shift numbers more quickly closer to home.

With its cramped quarters and high concentration of wealth, New York City has long had a large number of residents who own second homes. By cross-referencing property records, voter rolls and other data, MoveIndigo identified more than 40,000 people it believed to be Democrats or Democratic-leaning voters who owned property in the very districts their party needed to flip back.

The group found that few of these people realized that they could legally vote from them. Other states largely dissuade or outright prohibit their residents from voting anywhere except a primary residence or legally designated domicile; California, for example, cracked down this year on second-home owners trying to register to vote in Lake Tahoe.

But New York courts have made it clear that state election law allows it if the voter has “legitimate, significant and continuing attachments” to the second home. And the state’s move to liberalize voter registration and vote-by-mail rules has made it even easier in recent years.

MoveIndigo’s pitch is blunt: “Your second home could determine the next speaker of the House.”

The group has raised close to $450,000 to dispatch tens of thousands of letters, postcards and registration forms, relying on large checks from Tom Brokaw, the longtime NBC anchorman; the former New York City councilman David Yassky; and the Sixteen Thirty Fund, a dark money group that funds a slew of causes on the left. It has also partnered with local Democratic committees to promote its work.

“The opportunity is to cast a vote that could be a game changer at the ballot box in a place that you feel deeply about and have a long-term connection to,” said Ms. Weston, who in addition to casting her own ballot in the 19th District, helped found MoveIndigo with Charles Simon, a lawyer.

“Our role,” she added, “is to educate people that it’s legal and advantageous to do it.”

The push does not come without risks, especially for Democratic candidates already trying to rebut Republican accusations of elitism.

Adding more wealthy urbanites to the voter rolls in places like Long Island and the Hudson Valley could further stoke tensions that stretch back generations between full-time residents and the weekenders many blame for driving up housing costs, gentrifying their towns and setting statewide policy. The situation is particularly tense in the Hudson Valley, where an influx of full-time residents from Brooklyn and Manhattan is already reshaping the region in its image.

“It’s not enough that they are destroying New York City with their sanctuary city and pro-criminal policies,” said Matt Organ, Mr. Molinaro’s campaign manager. “Now Democrats are shipping these policies north, trying to hijack local priorities.”

Mr. Zerkin, the professor and an expert in conflict management, said he understood that concern, but he did not see himself as a cause for it. Though he and his wife now live primarily in Prospect Heights, Brooklyn, they have been spending time at their home in Woodstock since the 1980s, even living there full time for a decade. Like other second-home owners, they pay local taxes.

“We have some roots up there and friends, and even some political connections,” he said.

At the end of the day, though, Mr. Zerkin, a loyal Democrat, said his own calculation was much simpler.

“One’s vote up there matters a lot,” said Mr. Zerkin, who plans to cast his ballot for Representative Pat Ryan in New York’s 18th District. “That just seemed like a no-brainer.”

At one point, Mr. Baier wondered if the vice president considered Mr. Trump’s supporters “stupid.” (“I would never say that about the American people,” she replied.) At another point, he asked if she would apologize to the mother of a murdered 12-year-old Texas girl whose death is frequently invoked by Mr. Trump because two recent Venezuelan migrants were charged with the crime.

Mr. Baier’s aggressive demeanor was consistent with the kind of tough coverage of Ms. Harris that blankets Fox News’s daily programming. Lots of viewers were surely eager to hear how she would respond when confronted head-on.

Frequently, however, Mr. Baier did not give viewers that chance. Instead, looking frustrated, he cut off several of Ms. Harris’s answers after a few seconds. His first interruption came within the first half-minute of their exchange.

“May I please finish responding?” Ms. Harris asked at one point. “I’m in the middle of responding to the point you’re making, and I’d like to finish.”

A veteran interviewer, and the face of Fox News’s relatively nonpartisan 6 p.m. newscast, Mr. Baier was eager to shake Ms. Harris off her usual talking points, including playing a Trump campaign ad and asking her to respond.

That approach yielded dividends. Pressed on differences between herself and President Biden, Ms. Harris issued one of her clearest comments yet creating distance from the current administration: “My presidency will not be a continuation of Joe Biden’s presidency,” she said.

Still, in his eagerness to prevent a filibuster, Mr. Baier’s follow-up questions sometimes resembled rebuttals. Afterward, on a political roundtable with Fox commentators, Mr. Baier sounded a touch defensive. “I tried to redirect numerous times without interrupting too much, but at some point, you kind of have to redirect to get back in the game,” he said.

Any high-stakes interview is a high-wire act, and Mr. Baier later acknowledged that he had been frustrated before the encounter had even begun. He said the taping had been scheduled to start at 5 p.m., less than an hour before his newscast, and described what he called last-minute shifts by the Harris campaign.

“We were going to do 25 or 30 minutes. They came in and said, ‘Well, maybe 20,’ so it was already getting whittled down,” Mr. Baier said. “Then the vice president showed up about 5:15. We were pushing the envelope to be able to turn it around for the top of the 6 o’clock. So that’s how it started, and I could tell when we started talking that she was going to be tough to, you know, redirect, without me trying to interrupt.”

He said he eventually had “four people waving their hands” in the room, telling him to wrap up.

For weeks, Ms. Harris has faced criticism for avoiding adversarial meetings with journalists. (Mr. Trump has mostly done the same.) Agreeing to her first-ever Fox News interview was an effort by the Harris campaign to allay those concerns, and to share her viewpoints directly with an audience that may not often hear them.

Her aides declared themselves pleased with the results.

“We feel like we definitely achieved what we set out to achieve, in the sense that she was able to reach an audience that has probably been not exposed to the arguments she’s been making on the trail,” Brian Fallon, Ms. Harris’s campaign communications director, told reporters on Air Force Two. “She also got to show her toughness in standing tall against a hostile interviewer.”

Mr. Baier’s colleagues on Fox News praised his performance — “It was amazing that you were the interviewer,” Dana Perino said — while sounding far less impressed by Ms. Harris. The anchor Martha MacCallum deemed Ms. Harris’s answers “thin” and told Mr. Baier, her co-host on election nights, “You really asked the questions that a lot of Americans want answers to.”

Mr. Trump has refused to participate in another debate with Ms. Harris — even on Fox News, whose attempt to host a prime-time meeting of the candidates later this month was quickly rejected by the former president’s campaign.

For now, Mr. Baier’s grilling on Wednesday evening may have to stand in.

But Ms. Harris — giving her first interview on Fox News in an attempt to reach millions of voters, especially conservative-leaning women, who have probably not heard much of her message — largely steered the conversation in her preferred direction.

Here are six takeaways from the interview.

She broke with Biden (a little).

Ms. Harris made her clearest effort to separate herself from Mr. Biden after she was asked how her administration would be different.

“My presidency will not be a continuation of Joe Biden’s presidency,” she replied, adding that she represented a different generation of leadership and would address issues like housing and small businesses in different ways.

Republicans have seen Ms. Harris’s unwillingness to articulate differences from the unpopular president as a political gift. In an interview on ABC’s “The View” last week, she said there was “not a thing that comes to mind” when asked what she would have done differently from Mr. Biden.

Ms. Harris has engaged in an awkward dance with her boss, walking a line between being deferential to their administration’s accomplishments while trying to assert her own authority. Her answer on Wednesday was more rhetorical than substantive, but Mr. Biden may have given her the green light to be more aggressive going forward.

“Every president has to cut their own path. That’s what I did,” he said in a speech on Tuesday. “I was loyal to Barack Obama, but I cut my own path as president. That’s what Kamala’s going to do. She’s been loyal so far, but she’ll cut her own path.”

And when Mr. Baier slyly asked her when she had first noticed that “President Biden’s mental faculties appeared diminished,” she gave little ground, saying Mr. Biden was more than capable of leading the nation — unlike Mr. Trump, whom she called “unfit,” “unstable” and “dangerous.”

An aggressive Bret Baier pushed right-wing arguments.

From the outset, Mr. Baier seemed determined to knock Ms. Harris off her talking points — often by echoing those of Mr. Trump.

It took him less than 20 seconds to interrupt her for the first time. Ms. Harris, who is known as an effective practitioner of the filibuster, had hardly even begun to answer his opening question. That pattern continued for much of the interview.

Many of Mr. Baier’s questions seemed drawn from Mr. Trump’s own arguments. The Fox News anchor invoked the names of young women whom Mr. Trump frequently points to at his rallies as victims of undocumented immigrants and who are often cited on Fox programming. In fact, nearly half of the 26-minute interview was devoted to immigration and border security, issues seen as among Ms. Harris’s biggest weaknesses with undecided voters. Mr. Baier also suggested she was soft on Iran.

At one point, the Fox anchor came to Mr. Trump’s aid by showing a clip of him defending himself from criticisms of his “enemy from within” comments. (Ms. Harris quickly pushed back.) At another, Mr. Baier played a Trump campaign ad and asked Ms. Harris to respond.

It was a far cry from the friendlier confines of MSNBC, nonpolitical podcasts and local radio stations where Ms. Harris has given other interviews.

For Harris, the interview was largely meant to appeal to women …

The interview with Mr. Baier gave Ms. Harris access to a large audience of Republican women whom her campaign is trying to win over. Her advisers believe there is a sliver of conservative women who might be receptive to the character contrast she is trying to draw with Mr. Trump — or who are at least willing to hear her out.

Harris campaign officials believe that talking about the current landscape of abortion restrictions in the United States is a winning strategy with female voters, particularly liberal and liberal-leaning ones. But Mr. Baier did not bring up the issue, and the vice president did not guide him there.

Instead, both of them stayed focused on immigration and border security — a topic that, according to recent polls, is near the top of the list of concerns among female voters.

At several points, Mr. Baier asked the vice president if the families of women killed by undocumented immigrants were owed apologies from Ms. Harris and the Biden administration. He read off their names from a list, one by one, and played a clip from the mother of one of the victims, who blamed the administration’s border policies for the loss of her daughter.

At every turn, Ms. Harris paused to express her condolences. But she repeatedly redirected the conversation back to Mr. Trump’s work to sabotage a bipartisan border bill that would have amounted to the toughest restrictions in years, and pointed out that she was the only candidate running who had prosecuted criminals, including members of cartels.

“Let’s talk about what is happening right now with an individual who does not want to participate in solutions,” Ms. Harris said of Mr. Trump.

… and those women saw the vice president being interrupted repeatedly.

During this portion and others, the viewers Ms. Harris and her campaign are trying to appeal to also saw Mr. Baier repeatedly interrupt her as she tried to answer his questions.

The back-and-forth recalled how Matt Lauer talked over Hillary Clinton during a televised NBC News forum in 2016. Mr. Lauer was roundly criticized for being sexist.

“You have to let me finish, please,” Ms. Harris said at one point during the exchange on immigration. “I’m in the middle of responding to the point you’re raising, and I’d like to finish.”

As Mr. Baier’s interview with Ms. Harris progressed on Wednesday, he seemed more willing to let her have her say.

“We’re talking over each other,” he said shortly before it ended, though he had been doing so himself throughout their encounter. “I apologize.”

The interview showed the limits of her outreach to Republicans.

Ms. Harris frequently made points that Fox News viewers don’t often hear in their normal programming, saying that Mr. Trump was unfit to serve and pointing out the number of former officials in his administration who support her candidacy.

But the interview was a reminder that even as she talks in speeches about establishing a cross-party dialogue — and campaigns with Republicans like former Representatives Liz Cheney and Adam Kinzinger — the vast majority of the Republican Party and its media ecosystem view her with skepticism bordering on contempt and stand firmly behind their chosen candidate.

At the same time, some younger and progressive Democrats have watched warily as Ms. Harris goes to great lengths to court moderate and conservative voters — a general-election strategy that is common but still carries the risk of dampening liberal enthusiasm.

Mr. Trump’s campaign later trolled Ms. Harris by sharing the whole interview on social media, referring to it as his latest campaign ad.

Harris flipped a Trump transgender attack back on him.

Mr. Trump has tried to hammer Ms. Harris over her past support for prisoners receiving gender transition care. The Trump ad that Mr. Baier played for Ms. Harris — which many Republicans see as one of the former president’s most effective — said that she was in favor of “taxpayer-funded sex changes for prisoners” and “illegal aliens.”

Ms. Harris quickly sought to turn the tables.

“I will follow the law, and it’s a law that Donald Trump actually followed,” she said with a wide smile, pointing out that federal prison officials under Mr. Trump provided an array of gender-affirming treatments, including hormone therapy, for a small group of inmates who requested it during his term in office.

Transgender inmates are among the most vulnerable people in federal prisons, and have received significant protections from the courts.

“Frankly, that ad from the Trump campaign is a little bit of like throwing stones when you’re living in a glass house,” Ms. Harris said.

Reporting was contributed by Lisa Lerer , Reid J. Epstein , Glenn Thrush and Michael M. Grynbaum .

In a swift, 11-page ruling, Judge Thomas A. Cox found each of the rules passed by the Georgia State Election Board to be unlawful, violating Georgia state law, the Georgia State Constitution and the U.S. Constitution.

While the ruling amounted to a paragraph-by-paragraph rejection of months of rule-making by the State Election Board, it also struck a clear blow at the heart of many right-wing election policy goals, refuting arguments about election certification, hand counting and identification requirements as contrary to current law.

Since May, the Georgia State Election Board has been governed by a 3-2 right-wing majority, passing a host of rules that have been met with near universal opposition from local election officials. The secretary of state and attorney general, both Republicans, have warned the State Election Board that it was most likely exceeding its legal authority.

Judge Cox agreed.

“The S.E.B.’s authority can only extend to adopt rules and regulations to carry into effect a law already passed or otherwise administer and effectuate an existing enactment of the General Assembly,” Judge Cox wrote. “The rules at issue exceed or are in conflict with specific provisions of the Election Code. Thus, the challenged rules are unlawful and void.”

The lawsuit challenging the new election rules had been brought by a group of Republicans known as Eternal Vigilance Action, founded by former State Representative Scot Turner, a Republican, who praised the decision on Wednesday.

“This is a victory for the Constitution and the principle of separation of powers. Every conservative should see this as a win and significant pushback on an unelected board making law,” Mr. Turner said.

During the hearing, a lawyer for Eternal Vigilance Action argued that the State Election Board’s activity was an example of bureaucratic overreach, and not conservative government.

“Kind of like Napoleon, they put a crown on their head and say we are the emperors of the election,” said the lawyer, Christopher S. Anulewicz, referring to the state election board. “No. That is not the way that our system of government works.”

A spokesman for the State Election Board did not respond to a request for comment, and the prospect for an appeal of the judge’s ruling was unclear.

Dr. Janice Johnston, a Republican appointee to the election board, attended the hearing, as did Mike Coan, the board’s executive director. Cleta Mitchell, a Republican lawyer who played a key role in trying to overturn the 2020 election, attending the hearing via Zoom.

The chair of the Georgia Republican Party, Josh McKoon, called the judge’s decision “totally nonsensical” and “entirely wrongheaded” in a statement posted to social media. He said the party would appeal the ruling.

“These common sense rules will increase public confidence in elections administration,” Mr. McKoon said.

The decision from Judge Cox followed a ruling on Tuesday from the same court — but in a separate case — that blocked a rule passed by the State Election Board requiring the hand counting of ballot totals, pointing out that the rule would have taken effect midway through early voting. In his ruling on Wednesday, Judge Cox made similar note of the timing, writing that the rules were “passed approximately a month before early voting began.”

Judge Cox’s order was filed less than an hour after oral arguments were held over a pair of lawsuits also challenging new state election board rules. He directed the state to “immediately inform all state and local election officials that these rules are void and are not to be followed.”

In his ruling, Judge Cox repeatedly criticized the State Election Board for ignoring current law.

Regarding a rule requiring video surveillance of drop boxes after polls close, Judge Cox wrote that “the S.E.B. cannot by rule require something the General Assembly both did not legislate and specifically considered and declined to enact.”

Regarding a rule mandating hand counting of ballot totals to compare to machines, Judge Cox wrote, “This hand counting exercise is nowhere authorized by the General Assembly in the Election Code.”

And regarding a rule requiring that a photo ID or a signature be presented when delivering absentee ballots, Judge Cox wrote that the election board had “no authority to require such presentment as a condition of accepting and counting an otherwise properly delivered ballot.”

During the hearing on Wednesday, Judge Cox suggested the state’s election board could have instead asked elected politicians in the Georgia General Assembly to make the changes by passing new laws.

“Wouldn’t this have been a more appropriate vehicle to get these rules approved?” asked Judge Cox.