Until recent years, this was the practice. You grab them, log their names, and push them back through the fence. They could be back on the Mexican side of the border the very same day.
It is only in the last few years, say, under Obama, that suddenly a common fence climber could claim “asylum” status.
I would say that no one coming from the western hemisphere has a legitimate claim for asylum, and if they do, they are probably coming by air. If your government has put a contract on you personally, to assassinate you, and if we like you, then you have a claim. Otherwise, no.
So you’ll find a few political people on the run, say, from the Venezuelan government may qualify for asylum while they plan their return to the fight. A farmer from Honduras doesn’t. He doesn’t even merit a hearing.
Furthermore, we don’t need to be returning them to their country of origin. Mexico let them in, they are Mexico’s problem. Log their name and maybe a DNA sample if they don’t have valid ID. Push them back through the fence.
So this practice was just an “informal” practice? So why can’t Trump tell the border agents to go back to status quo and escort the Mexicans back?