How about instead of getting your knickers in a knot over "Amnesty"- for or against- you accept the fact (yes, FACT) that it can never be a crime to simply move to where you would rather be, as long as you don't trespass on private property in the process? No matter what words have been written or uttered to that effect.
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.