It could be fixed for the class with a class action. But the courts have also don't their damndest to make that hard too. Requiring a class action means that courts have the additional opportunity to say "nope that's not a valid class" (WalMart v Dukes being a rather famous example).
I read the Wikipedia article [0] and I would point out that a class defined by birth place would pass easily Scalia's test for commonality. So it seems that using a class action suit that includes every person born in US of un-documented parents would work. (But who knows what other tests the Supreme Court would come up now...)
