The number of children being moved on domestic flights is unclear. The New York Times contacted media offices of both the Department of Homeland Security and the agency under its jurisdiction, Immigration and Customs Enforcement, but only received the statement from Mr. Houlton in response.

The Times also talked to several flight attendants who said they had been disturbed by seeing Latino children on American Airlines flights, accompanied by government agents. But they declined to give their names, for fear of retaliation.

The topic has been a searing one for flight attendants. Sara Nelson is the international president of the Association of Flight Attendants-CWA, which represents 50,000 flight attendants at 20 airlines (American’s flight attendants are represented by a different union). In a statement, Ms. Nelson acknowledged that the topic was a heated one among members.

“This national discussion and response is being felt on the planes and discussed among crews,” Ms. Nelson said in a statement. “Some are struggling with the question of participating in a process that they feel deeply is immoral.”

For flight attendants, the mere possibility that they could be involved in a process that is increasing, rather than ameliorating, children’s fears runs counter to one of their core missions. They are regularly entrusted with the care of unaccompanied minors, and take pride in making them feel welcome, many say. Jill Sanders, 51, an American Airlines flight attendant with three decades of experience, said that they were also trained to recognize and report cases of children who might be victims of human trafficking.

“We all can tell at least one story of holding a child while their mother goes to the bathroom on the airplane,” she said, “and know what it’s like when their mother is gone for one minute and 30 seconds.”

Ms. Sanders said she had not personally seen any children who appeared to be migrants separated from their families. But she said she had read several accounts from her colleagues that had been posted on a private Facebook page.