Yeah, there's no guarantee that you'll capture the "Continental" culture (European/Asian/etc.) together with any given city.
But after you capture enough cities with that building surviving through the owner switch, you get that "Continental" culture on a permanent basis (or should do that).
Meaning, if your European Spanish pioneers keep assimilating Native Indian cities, eventually you WILL get the "South American" culture even in Madrid, lol.
Note, though, that "Culture" is still a resource like any other, so it only spreads the same way any other resource would spread.
As in, if you have a city that isn't a part of a currently enabled trading route system, it WON'T get the cultures built in your other cities - just like it WON'T get stuff like Cow or Silver either.
This is somewhat important, if you expect to, say, get Sioux into Moscow via Madrid - your Moscow might NOT be connected to Madrid by any trade routes for this to happen to begin with.
(The above means a case where Spain took over both Native Indians and Russia - both of the two conquered regions are not necessarily connected to your capital via trade routes, lol.)