Michelle Brané, the director of the migrant rights and justice program at the Women’s Refugee Commission, a research and advocacy organization, said that while groups like Southwest Key cannot act in a law enforcement capacity and lacked the authority to “tackle a kid and restrain them,” officials there must call the authorities when children leave the premises because they are technically in federal custody.

It is possible the children who leave simply want to “go find their parents,” she said, adding, “If children are running away, that raises questions about the care that’s being provided.”

Mr. Eller said the teenager left Casa Padre at 3 p.m. on Saturday and did not return. Staff members tried to keep the boy there, as they do with all children who want to leave, he said.

“It’s a conversation, a conversation that we’re here to help you, we’re here to help you get back with your family,” he said. “We can’t help you if you decide to leave.”

The teenager’s leaving was reported to the police in Brownsville, who did not respond to requests for comment on Sunday.

Federal officials said the boy had presented himself at the border alone. If the teenager is found, he will go back into the custody of the Department of Homeland Security and be referred again to the Office of Refugee Resettlement, officials said.

Tony Martinez, the mayor of Brownsville, said he could not recall a previous case of a child walking away from Casa Padre.