HIDALGO COUNTY, Texas—Sgt. Dan Broyles once had to battle through the spiky thicket of border vegetation here to find an immigrant illegally sneaking into the country.

But all he had to do on a recent day was to wait in plain sight along a dirt road, as a group of seven Salvadoran migrants, including a 6-year-old girl with a pink Hello Kitty backpack, deliberately walked up and surrendered to him a mile north of the Rio Grande.

"They're...