This is crazy talk. No countries are waiting until they see a prima facie case of harassment and then saying, "We've seen the facts and we're giving asylum!" Every country that could conceivably do that has weighed the damage to their relationship to the US and stayed well away.

I don't even think that the collection of Latin America would-be asylum-givers are very serious. In the most likely case, they are just grandstanding for their own reasons.