I can only write for Vista, at least as it was some years ago(till I managed to leave this outfit):
Proceeding base is usually in your contract(!!) which means officially it is not really easy to change. Why? For instance inside the EU your base will be then used for tax deductions, e.g. Italian base, Italian tax and social contributions and so on. You see already, not an easy thing to change bases.
Same goes for moving "outside" the EU - then it is your responsibility to follow tax laws(most just do that offshore thing which is quiet a dangerous gamble nowadays). I KNOW of some guys living in Asia(KUL if I remember right) and the company proceeded him in and out of there. BUT he was a big company man, checking if everyone is REALLY having the cheapest possible proceeding back home and such BS - so the company did him a return favour for making his collegues life miserable. I guess he is still there....
BUT - he was not a registed in Malaysia, had bank account and flat on his local girlfriends name(she stole some of his money as well). On his own he could not be based in KUL. Basically he was a tourist for the local authorities(as he left every month for duty there was no issue with that, immigration wise. All the rest was "fishy" at best).
As I know Vista they might not object IF you would move to a country that would reduce costs for them(less social contributions, NO tax, etc.) BUT it might be a big problem if you would need for instance to come back to the EU which would in turn, raise their costs again. I doubt they would be happy and cooperative.
So or so - you would be always on your own sorting out things with authorities if outside the EU(inside the EU just a few countries are allowed, e.g. Austria, Italy, I think Spain, UK, Germany and some of BeNELux, maybe Denmark).
License wise there should be no problem. You always work for Vista Malta(VJS, was VJT("Vista Malta") as long as the "original" Austrian Vistajet was still existing, then it moved back to VJS), so no need to change license authorities.
There is(was?) a US "Version" of Vista, they got some of the original aircraft ordered for VJS. Do not know(or care to be honest) if they are still around. Vista US is completely independent and as a VJS pilot you would never fly one of their aircraft(US registered).
Hope that could shed some light on things. However, I want to mention that again, this info is now a couple of years old as I left them as soon as I had a chance to do so(which should make you think ^^).