There were two main ways in which these societies were able to sustain their cultural practices. First, they did it by adapting their cultural practices to the institutions of Europeans who ruled them. Second, they were able to do it because they were given some degree of space and they took full advantage of that space.
The Europeans gave these societies space to exist. There were relatively few Europeans compared to Mesoamericans and Africans and so supervision was not constant or complete. When left to themselves, these people resisted their colonial masters by creating their own communities with their own cultural practices.
Because they were not completely left alone, the societies had to adapt themselves to European ways. One way in which this can be seen is religion. Both Mesoamericans and Africans fused their own religious practices with Christian practices. They found ways to locate their deities and practices within acceptable Christian practices (for example, by associating certain spirits with certain saints).
In these ways, these societies were able to sustain at least significant aspects of their own cultural practices.