Doom, Disaster, and Amnesty!
Instead of hacking in futility at those branches of evil- the ones labeled "they'll overwhelm the social services" and "they'll vote for more collectivism" or "they'll take our jobs"- strike the root.
It is a disastrous idea to set up a State, or to allow one to continue. States are always, and invariably, ONLY used to violate the rights of individuals. None of the above "issues" would be imaginable without The State making them possible.
That is the source of any problem with "Amnesty". Face it, accept it, and deal with it however you see fit.