TAPACHULA, Mexico—An estimated 5,000 Honduran migrants—some on foot carrying children and some hitching rides on trucks—set out Monday from southern Mexico on a grueling journey toward the U.S. border, as President Trump threatened to end or cut foreign aid to Central American countries for failing to stop the caravan.

The caravan’s 11-day march through Honduras and Guatemala and into Mexico has fueled a fresh political rift between Mr. Trump and those countries’ leaders just two weeks before U.S. midterm elections.

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