“They’re completely cut off from the outside world,” she said. “And officials are saying they’re free.”

In some cases, parents who recovered their children were dismayed by what they found. At the airport in McAllen, Tex., a migrant mother who identified herself only as Roxana, 23, was waiting to leave with her son Eduardo, 9, who had just been returned to her after being held in New York. But the boy had a black eye and a heavy cough.

“They gave him back sick and beaten and traumatized,” Roxana said.

Last-minute logistical planning to meet the deadline appeared to have led to some mistakes. Some children have been sent to the wrong facilities, according to a government official who spoke on condition of anonymity because they were not authorized to discuss the reunification efforts.

On Tuesday, two siblings, 9 and 14, were abruptly flown from New York to be reunited with their mother in the Southwest — but the mother may have already been deported, according to the children’s lawyer, Priya Konings. On Wednesday, a case worker was in “panic mode” trying to resolve the situation.

The hours and days after reunification have come with their own challenges, which have largely been absorbed by aid groups tapped by the government to provide emergency shelter and food, as well as transport. While it is not typical for the government to support migrants once they are released from federal custody, the separation policy left many families far from the border, where a long-established network of shelters and volunteer organizations have traditionally provided support.

Danielle Bernard, a spokeswoman for Lutheran Immigration and Refugee Service, one of two organizations working with the government on family reunifications, said her group, too, was feeling the time crunch of the court order.

Ms. Bernard said that the organization’s two offices coordinating efforts in Arizona and New Mexico were given only a few days to prepare to receive about 300 families each — or at least 1200 people in all.