Election interference cases against President-elect Donald J. Trump and his allies are moving forward in Georgia and Arizona, but recent complications in each case have fueled speculation about whether the prosecutions are in more fragile shape than they were before Mr. Trump won the election.

While nobody expects Mr. Trump himself to be tried while he is in office, dozens of his former aides and allies are still being prosecuted in at least four states. Some of the most prominent Trump associates, including Rudolph W. Giuliani, Mr. Trump’s former personal lawyer, and Mark Meadows, the former White House chief of staff, are facing charges in both Georgia and Arizona; Georgia is the only state in which Mr. Trump has been indicted.

But the cases are a long way from trial. This week, the Georgia Court of Appeals abruptly canceled oral arguments on whether the prosecutor leading the case in that state, Fani T. Willis, the district attorney of Fulton County, should be disqualified. A lower court judge had ruled against the defense’s effort to disqualify Ms. Willis, stemming from revelations that she had a romantic relationship with the outside lawyer her office hired to run the Trump prosecution.

The oral arguments had been scheduled for Dec. 5. The court offered no explanation for the move.

The appeals court clerk said in an interview on Tuesday that the cancellation would most likely not affect the mid-March deadline by which the court has to rule on the disqualification matter. While the judges have not said what drove their decision to call off oral arguments, the clerk said it might just signal that the judges decided to rely completely on legal briefs from the numerous defense lawyers.

“The information we got is that everybody wanted to speak,” said Christina Smith, the clerk of the Court of Appeals. “You have 60 minutes of argument and most of the eight lawyers wanted to argue. It would have been a lot of transitioning.”

Mr. Trump’s main lawyer in Georgia, Steven Sadow, declined to comment, as did Ms. Willis’s office.

The development in Georgia came on the heels of a decision last week by the presiding judge in a similar case in Arizona to recuse himself. Defendants in that case had called for the judge, Bruce Cohen, to be disqualified after it surfaced that he had circulated an email to judicial colleagues saying that his “blood boiled” after Mr. Trump reposted a crude joke about Vice President Kamala Harris and Hillary Clinton, and expressing anger about offensive statements made by some Trump supporters about Ms. Harris.

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