SAN PEDRO SULA, Honduras—On this city’s violent outskirts, where families live in tin-roofed shacks on streets menaced by gangs, the word is out about one of the biggest factors that can determine their fate in the U.S. immigration system: children.

The number of Central American families with children arriving at the U.S. border seeking asylum has surged in recent years—and they keep coming, as more migrant caravans make their way through Mexico.

Families...