This. Mexico offered and they declined.From a story I read a few days ago, Mexico has granted asylum to about 3000 of the migrants. The rest were just insistent on going to the U.S. and not staying in Mexico.
International law gives them no right to ask for asylum past the first country they reach after fleeing their own (all countries have to give it if they're the first country). Our laws are more generous, and give them the opportunity, but we're under no obligation.