Over the last three months, about 107,000 family members were caught at the border and then released, with ankle monitoring devices or check-in schedules and the often vain hope that they will show up for their court hearings and deportation.

Worse yet, officials say, they have had to pull deportation officers off duty in prisons and jails, where they were arresting criminal migrants, and deployed them to the border to help the Border Patrol release all the families.

“The current crisis that is occurring at the southwest border, the numbers that we collectively as a nation are seeing … is absolutely unprecedented,” said Nathalie R. Asher, acting executive associate director for Immigration and Customs Enforcement’s deportation division.