The President has it within his power to legally suspend the automatic granting of asylum to anyone who illegally crosses our borders.
His absolute authority to do so is already encoded into law, and was recently reaffirmed by the Supreme Court in the case of Trump vs Hawaii.
He can stop the horde in its tracks with one executive order.
The strategy is to ask for asylum at the crossing points. Heretofore it has been put into play for surreptitious crossers who get caught and say the magic words. Trump has deported some of those. If the caravaners, who have been trained as to what to say and how to say it will probably follow the procedure.