Migrants claiming asylum can be allowed into the U.S. Here's how it works

Yajaíra Peñaloza (left) and Marian Araujo pose with their children while waiting for their ride at the Casa Alitas shelter in Tucson, Ariz., on March 26. (Ash Ponders for NPR)

The heart of the debate over the U.S.-Mexico border is illegal immigration.

Yet that decades-old issue is complicated by hundreds of thousands of asylum-seekers claiming, under a process allowed by U.S. law, that they fear returning to their home countries.