Dr. Allen Keller, the director of the Bellevue/N.Y.U. Program for Survivors of Torture, who has flown to Mexico to meet with caravan members, called the situation “a refugee crisis.”

“These individuals are not here by choice,” he said. “In fact, they believe they have no choice.”

Melissa Guzmán, 33, from Honduras, was one of those who turned up early Saturday to put her name on the interview waiting list. She had joined the caravan with her daughter, Laura, 6, and son, Mynor, 11. Whatever the decision, she said, she wasn’t going back to Honduras.

As she walked back to the shelter with her cousin Yeni Palma, 37, and Ms. Palma’s 13-year-old daughter, Yesselin, Ms. Guzmán seemed to perk up at their surroundings. Vendors were setting up a street market, and the women gazed at the secondhand appliances.

Ms. Palma has begun to consider staying in Tijuana if she can’t reach the United States. Pressing her hand down on a bed at the street market to test the springs, she said, “It’s been a long time.”

Yet some of the migrants insisted that they will get to the United States, come what may.

José Adan Núñez, 24, was ready to take his chances to get across the border. After only a few days in the shelter, he declared, “If I die on the way, at least I will have fought for something.”