When the “Remain in Mexico” program was initially enjoined by a federal judge in California in April of last year, migrants who had been in court on that day ended up spending more than two weeks in government holding cells while officials decided how to proceed. When the injunction was stayed, the migrants were returned to Mexico, allowed to enter the United States only for their court hearings.

Government officials moved quickly to reverse the decision. It appeared clear that, whatever the immediate outcome, the issue would ultimately be decided by the United States Supreme Court.

Hundreds of asylum seekers who have been returned to Mexico have since given up their claims, accepting free transportation provided by the American government and the United Nations back to their homes in Central America. But others have vowed to continue with their cases.

Yoleydi Gonzalez Jimenez, 26, arrived from Cuba with her husband in the Mexican city of Matamoros in September and has been living in a tent encampment at the end of an international bridge into the United States ever since. With little access to public bathrooms, the camp smells of human waste.

Ms. Gonzalez Jimenez wears socks with her flip-flops to keep warm. A donated air mattress covered with pink and purple sheets fills the tent that has become the couple’s home. Their few possessions are stacked on top and become soaked with water that seeps inside when it rains.

“I can’t give up after all the time I’ve been waiting here, even though I feel like I’m going to die,” she said after a court hearing in December. Her next hearing was scheduled in Brownsville, Texas. Until then, she was told, she would have to go back to Mexico.

Reporting was contributed by Max Rivlin-Nadler, Manny Fernandez, Zolan Kanno-Youngs and Kirk Semple.