Maybe they thought their kids were at less risk than they would be staying in their own country?
So you're in favour of taking in an entire population?

do you not think you have a responsibility to at least try to feed and wash them?
Yes, no argument here.

so they don't have to face whatever horrors exist in their home country
Why do only the ones that show up at the border have these problems? And if they don't, why take in the ones who try to cross illegally at the expense of those who attempt to do so lawfully?

Better yet would be to follow international law
International law does not dictate that a nation must admit migrants.

France goes under in a civil war.
Are Mexico in a state of civil war? Or are you referring to the people who have fled various parts of Central and South America, and travelled through a peaceful Mexico, to get to USA?

Let's do this a different way... the UK becomes embroiled in civil war. Do I think I have the right to waltz to whichever country I can get to? No I don't. At the moment, I have the right to go to France, and French people have the right to come here. But let's fast forward a year or two and assume we're out of the EU, and we no longer have freedom of movement. I no longer have the right to go anywhere else.That might not stop me trying, but if I got politely told to fuck off, I wouldn't think I was being treated unfairly. I mean, why should I be allowed in but not the rest of the population? What makes me special? In fact, if I turn up at the border with a backpack, I'm less special than the people who actually apply for asylum before arriving.