The Trump administration plans to start pulling back on a controversial plank of U.S. immigration policy in a busy border region, saying Tuesday it will stop sending some migrant families who illegally cross the border in Texas’ Rio Grande Valley to jail.

Starting this week, hundreds of families caught each day in that area are being released by Border Patrol agents, instead of being handed over to U.S. Immigration and Customs Enforcement for potentially longer detention, government officials said. The exact number will depend...