WASHINGTON—The Trump administration took another small step this week in implementing a policy requiring asylum-seeking Central Americans to remain in Mexico while their claims are adjudicated.

The policy is now being applied to asylum seekers at the Calexico port of entry, Department of Homeland Security officials said Tuesday, as well as the San Ysidro port of entry where it began and other parts of the border near San Diego where people are crossing the border illegally.

The...