The same remote desert border crossing used by a 7-year-old girl who recently died in government custody is seeing a surge in families and children that the Border Patrol is unequipped to handle, its chief said Tuesday.

Jakelin Caal’s death, which has drawn criticism and heated up rhetoric around U.S. immigration policy, comes amid a problem current and former Border Patrol agents say has been growing for years: An agency designed to deter and deport adults traveling alone to find work now spends most of its time dealing with...