Some of Donald J. Trump’s co-defendants in the election interference case in Georgia began turning themselves in on Tuesday, while others tried to get the sprawling criminal case moved out of state court and into federal court.

Jeffrey Clark, a former Justice Department official, and David Shafer, the former head of the Georgia Republican Party, each filed motions on Tuesday asking to have the case moved to federal court, just as Mark Meadows, the former White House chief of staff, did last week. Those motions lay the groundwork for what will be the first major legal fight in the case, which was filed in Superior Court in Atlanta last week.

Most of the defendants, including Mr. Trump, plan to turn themselves in this week, as ordered by Fani T. Willis, the district attorney of Fulton County, Ga., who is leading the investigation. But Mr. Clark filed a request on Tuesday for an emergency stay, in a bid to avoid turning himself in at the notorious Atlanta jail where the defendants are being processed. Mr. Meadows made a similar request later in the day.

Another prominent Trump ally, John Eastman, turned himself in on Tuesday and was booked at the jail. Mr. Eastman, a chief architect of Mr. Trump’s effort to reverse his 2020 election loss, said in a statement that the indictment “represents a crossing of the Rubicon for our country, implicating the fundamental First Amendment right to petition the government for redress of grievances.”

“As troubling,” the statement continued, “it targets attorneys for their zealous advocacy on behalf of their clients, something attorneys are ethically bound to provide.”

State criminal prosecutions can be removed to federal court under a federal statute that allows for such a change of venue if the case involves federal officials and pertains to actions taken “under color” of their office. The term refers to things done in an official capacity or as part of official duties.

Last month, a federal judge rejected Mr. Trump’s efforts to have another state criminal case against him removed to federal court. That case, in New York, centers on Mr. Trump’s role in hush-money payments to an actress in pornographic films. In his order, Judge Alvin K. Hellerstein wrote: “Hush money paid to an adult film star is not related to a president’s official acts.”

If the motions for removal of the Georgia case are successful, the defendants would probably then argue in federal court that they should not be charged for state crimes, and would base that argument on the Supremacy Clause of the U.S. Constitution, which says that federal laws generally take precedence over state laws.

Three of the 19 defendants charged by Ms. Willis were federal officials at the time of the election: Mr. Trump, Mr. Clark and Mr. Meadows. Mr. Meadows filed a motion on Friday asking a judge to dismiss the charges against him, based on the Supremacy Clause; on Tuesday, he asked a federal court to either “immediately” permit removal or allow him to avoid arrest in Fulton County until a hearing on his removal request early next week.

Mr. Meadows included in his filing an email to his lawyer from Ms. Willis, in which she rejected the idea of letting Mr. Meadows delay turning himself in. “Your client is no different than any other criminal defendant in this jurisdiction,” she wrote, describing the nearly two weeks she had given Mr. Meadows and his co-defendants to surrender as “a tremendous courtesy.”

Mr. Clark’s lawyer, Harry W. MacDougald, argued in a legal filing that his client “was a high-ranking U.S. Justice Department official at all relevant times applicable to the Fulton County Action,” adding that “the allegations therein relate directly to his work at the Justice Department as well as with the former President of the United States.”

The Fulton County jail is under investigation by Mr. Clark’s former employer, the Department of Justice, for allegations of “unsafe, unsanitary living conditions,” the use of “excessive force and violence,” and a failure to provide adequate health services to detainees.

Mr. MacDougald wrote in a separate motion that “Mr. Clark should not be forced to enter and stay in that jail for any reason.”

Judge Steve C. Jones of United States District Court in Atlanta ordered Ms. Willis’s office on Tuesday to reply to the motion for an emergency stay by Wednesday at 3 p.m.

The racketeering indictment obtained by Ms. Willis laid out eight ways the defendants are accused of trying, as part of a “criminal enterprise,” to reverse the results of the 2020 presidential election: by lying to the Georgia legislature, lying to state officials, enlisting or acting as fake pro-Trump electors to circumvent the popular vote, harassing election workers, soliciting Justice Department officials, soliciting Vice President Mike Pence, breaching voting machines and engaging in a cover-up.

Mr. Shafer was never a federal official, but he was one of 16 Georgia Republicans who sought to cast false Electoral College votes in favor of Mr. Trump on Dec. 14, 2020, after Joseph R. Biden’s victory in the state had been certified by state officials.

Mr. Shafer’s lawyers made a novel legal argument that their client had the right to remove the case to federal court because, as a “presidential elector,” he was “acting under the authority of the Constitution and the Electoral Count Act,” and was doing so “at the direction of the President and other federal officers.”

Removal to federal court would broaden the jury pool for a potential trial. Instead of drawing jurors just from Fulton County, where 26 percent of voters chose Mr. Trump in the 2020 election, they would be drawn from a 10-county region that includes Fulton along with more suburban and exurban counties where Mr. Trump won just under 34 percent of the vote.

A number of legal experts say that moving the Georgia case to federal court would not allow Mr. Trump to pardon himself, if re-elected, after a conviction in the case. The Constitution grants presidents the power to pardon “offenses against the United States,” but the crimes charged in the Georgia case, wherever they are tried, are offenses against the state of Georgia, said Anthony Michael Kreis, a constitutional law expert at Georgia State University.

Mr. Meadows, in his filing last week, said that actions he took — including his involvement in a January 2021 phone call between Mr. Trump and the Georgia secretary of state, Brad Raffensperger, that was a focus of the Georgia investigation — fell within the scope of his duties as chief of staff to the president.

A hearing on Mr. Meadows’ request is scheduled for Monday at 10 a.m. before Judge Jones, who was nominated to the bench by President Barack Obama.

The filings in federal court from Mr. Shafer and Mr. Clark offer some of the most forceful pushback to date against Ms. Willis and her prosecution.

Mr. Shafer’s filing asks the federal courts to “bar” the case, which, Mr. Shafer says, “seeks to criminalize the actions of persons acting pursuant to federal authority to achieve the purposes of the national government.”

The filing continues: “Neither the state of Georgia nor any of its localities has the authority to prosecute Mr. Shafer for these actions, and this court should exercise its clear authority to correct this injustice and halt this unlawful and unconstitutional attempted prosecution now.”

Mr. Shafer’s bond was set at $75,000 on Tuesday. The bonds for Mr. Eastman and for Jenna Ellis, another of Mr. Trump’s former lawyers, were set at $100,000, and for Michael Roman, a former Trump campaign aide, at $50,000.