The Trump administration moved to dramatically limit Central American migrants’ ability to seek asylum at the U.S. border with Mexico, an escalation of the president’s push to stem the flood of border crossers that is severely straining the U.S. immigration system.

Under the rule published online on Monday, with limited exceptions, migrants who pass through another country first must seek asylum there rather than at the U.S. border, where they will be ineligible to do so. The vast majority of asylum seekers at the U.S.-Mexico...