But it has turned on novel issues about when diplomatic immunity ends and an exception to the general rule that anyone born on American soil is automatically a citizen. Ms. Muthana’s father, Ahmed Ali Muthana, was a United Nations diplomat from Yemen. Children of people with diplomatic immunity are not automatically granted citizenship despite being born in the United States. But Mr. Muthana was discharged from his diplomatic post shortly before she was born in 1994.

The United States government, however, said it was notified about that change in early 1995, after she was born, and stopped considering him covered by diplomatic immunity only then. The gap raised the question of whether his diplomatic status expired as a matter of law before or after her birth.

Mr. Muthana, who stayed in the United States, later applied for naturalized citizenship for his older children, who had been born abroad, but did not apply for Ms. Muthana because the government had led him to believe she was already a citizen — including by issuing her a passport, his lawyers have said.

Judge Walton in March rejected a request by Mr. Muthana to speed up consideration of the case. This month, Mr. Muthana filed a new such motion in light of the “upheaval” that followed Mr. Trump’s recent announcement of the withdrawal of United States troops from the area, which precipitated the fighting between Turkey and the Kurds.

The motion said that Ms. Muthana had been moved to a different refugee camp, known as Roj, after receiving death threats at the first one where she was held, Al Hol, from Islamic State loyalists for having repudiated the group. It also said her son, identified in court documents only as John Doe, was sickly, and that their lives were in danger.