Crossing the border illegally (for now) is a civil misdemeanor and seeking asylum is legal under US and international law.

> Does the country they are fleeing have no culpability?

The US definitely has some culpability for making other places harder to live in.

https://en.wikipedia.org/wiki/United_States_involvement_in_r...

> Why are they leaving?

They're not doing it just for funsies.