That is also already happening.
Or, even worse, subject them to the terrible torture of "filling asylum paperwork."
[1] https://www.bbc.com/news/world-africa-66157259
(</sarcasm>)
When that becomes billions knocking at the door, even the most liberal lefty in the west will say "nope, can't do it".
Think about the US/Mexico walls. Now think of that scene from World War Z where the zombies storm Jerusalem.
It's going to be genocide, it won't be pretty at all.
https://www.texastribune.org/2023/01/19/national-guard-migra...
https://www.npr.org/2023/04/22/1171450334/two-men-shot-at-a-...
In Europe it's not shooting, but the treatments are called "bordering on torture" (by pro-migrants ONG, naturelle) https://www.liberation.fr/international/europe/une-violence-...
We have no way to know if this is going to end with massive shootouts at borders. The political costs of "just build makeshift camps, keep them fed and out of sights, eventually send them to another country, and let most people ignore them" is still lower than "handle a viral video of your army massively shooting migrants".