VILLA DEL ROSARIO, Colombia—To escape Venezuela on a recent day, nine migrants staggered through a thicket of elephant grass, waded the Tachira River with luggage over their heads and scrambled past a farm to reach this backwater crowded with their countrymen.

Like countless others, they had only a few dollars worth of Colombian pesos and a vague plan: Reach a big Colombian city or get as far as Peru, on foot if necessary. First, they had to navigate the clandestine trails of a lawless, sometimes violent border, which growing...