Essentially 'yes'. When 2 or 3 border agents come across groups of hundreds, are they expected to handcuff and transport them all ? They have been schooled to utter the magic 'amnesty, i'm afraid' and the agents have to get them to an area where they can get their first interview. In a real world, illegal entry and capture would mean an immediate turn around and forced back over the border. This of course cannot happen with the abba cadabra 'amnesty' spell.
I think you mean the magic word...asylum...but, whatever.