There is one other way that culture can change. In non-European provinces that contain colonial cities (1000-4999 pop), whenever nationalism ends, the colonial city's culture will change to the owner's primary state culture. This effect only works if the settlement culture is different from the base provincial culture. Note that nationalism usually lasts 30 years; you can see it via the increase in revolt risk. However, when a country gains ownership of one of its core provinces, nationalism ends immediately (and the culture changes if other conditions apply). Similarly in the case of a colonial city defection, there is no nationalism, and therefore there is immediate culture change.
Again, there are two very substantial limits on this effect: the colonial city must be non-European, and it must not match culture to the base provincial culture (which you can't see). Base provincial culture usually matches the settlement culture for settled provinces as they are in the Grand Campaign scenario. (A few exceptions exist, including the Canary Islands.) Thus, for example, China cannot easily expand into the Changatai Khanate and convert its many small cities into Han culture, because those cities are Mongol cities on provinces with Mongol base culture.