I think the whole immigration idea is lame for this game, unless they're willing to remake America out of immigrants. A civ can form out of immigrants more often than it can out of an indigenous nationality. Just look at all the countries in the Americas, Australia, New Zealand, South Africa, etc.
So.
To immitate the historic processes, countries with a large population of undesirables - unhappies, religious and ethnic minorities, criminals, etc. - should resettle such ppl in far away lands to get benefits in core cities. Naturally, such distant lands will some day separate and form new civs - it's guaranteed. Just like some wonders expire so will large empires, especially those stretching across the oceans, will break up after certain techs are discovered. This doesn't mean you don't have to develop such cities, development brings you stability, revenue, and postpones separation.
Furthermore, at the time of separation, you have the choice to choose which part you will control from that point. Also, both civs will most likely speak the same language, so there is a big chance for cultural friendship in the future.
The immigrants themselves should just get up and leave from towns that don't have enough food to grow or jobs (high corruption?) and automatically join a successful city in your civ or another civ. You will just get a message about this event.