Will converted civilizations actively spread your religion in Civ 6?

Yes and no.

When your cities get taken over, and you then spawn a missionary, you'll see that it's a missionary for the dominant religion in the city that spawned it.

The same if true for the AI when you convert their cities. Religious pressure will emanate from the AI's cities and it will spread.

However, it makes no sense that the AI would keep spreading the foreign religion, as they would then be helping you winning your religious victory which causes them to lose.

Yes, religion will emanate. No, the AI won't help you win the religious victory.