HUIXTLA, Mexico—A caravan of thousands of mostly Honduran migrants collapsed makeshift tents in this southern Mexican town and set off before dawn Wednesday on their long journey to the U.S., as thousands more of their countrymen made their way across Guatemala with the same goal.

The migrants are part of a surge in migration from Honduras in recent weeks as growing numbers of people join multiple caravans that allow them to travel safely together and avoid paying smugglers’ fees.

The...