Vice President Kamala Harris attacked former President Donald J. Trump on Thursday for claiming at a rally that he would protect American women “whether the women like it or not,” trying to elevate her opponent’s track record with female voters even as the Trump campaign lashed out at the billionaire Mark Cuban, a top Harris surrogate, for insulting the intelligence of women close to the former president.

It was another hairpin turn that took the presidential race from literal trash talk to gender issues in its closing stage, with both candidates trying to inflict political wounds that will take days to heal as Americans cast their votes.

Ms. Harris, speaking from Wisconsin on Thursday morning before leaving for campaign stops in the West, said that Mr. Trump’s comments, made the evening earlier at a rally near Green Bay, constituted a “very offensive” message to all Americans. Within minutes, the Trump campaign fired back: “Why does Kamala Harris take issue with President Trump wanting to protect women, men, and children from migrant crime and foreign adversaries?” Karoline Leavitt, a campaign spokeswoman, said in a statement.

On Wednesday, Mr. Trump had rolled into a Green Bay-area rally sitting in the passenger seat of a garbage truck and tried to tie Ms. Harris to comments made this week by President Biden, who appeared to call the Republican nominee’s supporters “garbage” as he criticized a comic at Madison Square Garden who, days earlier at a Trump rally, had disparaged Puerto Rico as “a floating island of garbage.”

But after Mr. Trump told the crowd that his advisers had urged him to stop using a well-worn rally line about his desire to protect women, saying they had called it “inappropriate,” the Harris campaign saw an opportunity to throw the focus of a race that has been divided along gender lines squarely back onto her opponent.

“This is the same man who said women should be punished for their choices,” Ms. Harris said at her rally in Phoenix on Thursday afternoon, after repeating Mr. Trump’s comment to a chorus of boos from the crowd, which the campaign said numbered over 7,000. “He simply does not respect the freedom of women or the intelligence of women to know what’s in their own best interest and make decisions accordingly. But we trust women.”

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