MATÍAS ROMERO, Mexico — The caravan of Central American migrants that President Trump has portrayed as an emblem of flawed immigration policy in the United States began to splinter on Thursday, as hundreds of people departed this rural town aboard buses and on foot, abandoning the decrepit municipal sports complex where their journey had stalled for five nights.

But the movement’s organizers insisted that the caravan would regroup in the Mexican city of Puebla in the next few days before continuing on to Mexico City, where it would officially end after meetings with government officials and possibly a street protest.

Mr. Trump has been fixated on the caravan this week, making it a recurring motif in his tweets, warning that it represented a grave threat to the sovereignty of the United States, and using it as a justification for sending the military to reinforce the southwest border.