This is not difficult.
If you break the law then you lose your right to remain in the USA. This includes the extended families. I'm not talking about a a tiny misdemeanour or simple traffic ticket.
If you are an illegal undocumented, you have never had that right to stay. This includes the extended families.
If you come from a dangerous country and gained asylum or a legal immigration status in the USA, then you should have thought about that before you broke the law.
If you enter without prior approval then you can of course seek asylum. Decision is made with 2 weeks. Yes or no. You have one attempt at an appeal, within 1 week, decision on that within 1 week.
If you are granted asylum then remember your stay is at the discretion of the President. Break the law? You know what will happen, it cannot be a surprise.
Same goes for the UK, or any other country that has accepts legal immigrants or asylum seekers.
I repeat, this is not difficult.