Religion + Ideological Pressure + Culture Swap. ^^
You spread your Affinity with trade routes, your spies and the new "Zealot" (Missionary) unit. Pressure causes unhealthiness when your affinity is not dominant in your city and if you lose too much/are too low for too long that city changes owner to the one who's pressuring your city. At the same time every city you have pressure on will give you points toward your affinity -> that's how you level it, the tech tree only gives buildings that increase the pressure your cities can produce (and strong, but expensive to maintain buildings - on other leaf techs, so not every game plays out exactly the same). The AI will of course hate you for spreading your affinity into their lands, enforcing conflicts (and ideally wars if they realize they can't withstand your Affinity Pressure) and thus getting rid of the very passive gameplay in the later half of the game.