james, your points and concerns are spot on, imo. others have already expressed some of what i would, so i'll just let my thoughts roll.
the fact that living beings (not just humans) are consistently and horrifically oppressed, controlled, enslaved, broken, destroyed, etc, is undeniable in today's world. how to deal with that when you are a thinking, feeling individual is... a dilemma for most, to put it mildly. i think a good part of that dilemma is the disconnect between what individuals think/feel/want/are capable of, and the ubiquity/power of authoritarian institutions that exist and operate in almost complete opposition to individual autonomy.
whether it is a nation/state or a massive corporation makes no difference to me, other than possibly strategy and tactics. borders and immigration are just aspects of authoritarian control tied to the concept of the nation/state. if you haven't read it, i suggest "the continuing appeal of nationalism" by fredy perlman (http://theanarchistlibrary.org/library/fredy-perlman-the-continuing-appeal-of-nationalism).
i doubt there is a single "best" way to help others that want and ask for it. each situation needs to be addressed in its own context. but i guess when you talk about toppling a massive corporation or government (again, i see all gov as dictatorships to varying degrees), you are talking about an "enemy" that has virtually infinite resources and support.
my approach would typically be to try to address the problem at its root. to me that points to civilization and mass society. but most people seem to see those as necessary, if not positively desirable. and so i would look at what is possible in the particular context being addressed. the idea that anyone could create a "safe space" - especially in distant locations - is questionable at best. helping them leave that area may be a useful approach. but as you point out, allowing people to leave does not make them safe, it merely removes them from one set of oppressors. and places them closer to others.
i do not see a viable solution at the macro level. at the level of individuals and their relationships, i think solutions need to come from them, based on their context (existing and desired).
bottom line: if the problem can only be seen in the context of nations and borders, then the solution is likely to be based on those same concepts. which is, needless to say, not anarchistic in any sense i am interested in. if, on the other hand, individuals and affinity groups decide to take direct action in ways that can circumvent (or, ideally, destroy) the power of the state, i am all for it.
not a particularly useful response, i know. but i'm glad to see someone new here engaging with these issues.