forced asylum

The current system will have these folks cycle through the justice system. A justice system that will still hold them against their will and treat them poorly anyway. For those people that need to be held (a very small amount, like you said), it will be better to hold them in a jail/prison that promotes rehabilitation and will treat them compassionately of they continue to fail to be rehabilitated. Very unlike the current jails/prisons in the United States.

what are they supposed to do?

What should they do? Don’t make it the disadvantaged people’s problem. Make it the problem of the people that failed these disadvantaged folks.

Put pressure on the government to fix the lack of services. (Or if they’re already part of the government, the correct people in government, like the governor for state care.)

We can’t keep passing the buck.

Again, I understand it is uncomfortable to fight against an unjust system. It’s not pretty. But the alternative is letting down people that need support in favor of people that would as soon grind us up next.