If you are going to try to culture flip them, add towns as close to their borders as possible, and rush temples in those towns. The more you "squeeze" their sphere of influence with yours, the better the chance of a culture flip. Also very important is how high your culture is compared to theirs in the F8 culture graph - the better the ratio there, the better the chance of getting them to flip. But it still seems to include a large random factor and be hard to predict.
Are you much stronger than the other Civ in terms of "power" in the F8 histograph? If you are much stronger than them, and their towns in your area are still quite small, there's a chance you can convince them to just GIVE them to you, or at least some of them. To do that, talk to them, click on "active" to see existing deals, and click on peace to renogotiate peace. Then try to add the cities you want (note their names before coming to this screen) to the deal and see if they will go for it. But a couple of warnings about this approach:
1) It can backfire if they want something from you for peace. If you are willing to experiment and reload if it goes wrong, no harm done. Otherwise, don't try this unless you are a fair bit stronger.
2) You'll be committed to peace for 20 turns after doing this unless you don't mind a black mark on your reputation. So you might not want to do this and then attack them afterwards anyway. (But I would
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:suicide: OR 
) and I feel that I want to avoid a full-scale onslaught right now :die!:. How many turns does it take to culture flip them?? My cities have MASSIVE culture, and so it's only a matter of time
