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.

Mvemba a Nzinga will actively spread foreign Religion to his own cities using charges from the Apostles he gets for free when completing certain buildings on Religious cities.

He will also, with the same Apostles, attack passing units of other Religions, occasionally killing them and converting nearby cities with the area effect of lethal Theological Combat, regardless of who owns the cities.

Other than Mvemba, I don't think I ever saw a single Religious unit of a Religion not founded by the unit's owner. I have played for hundreds of hours over several difficulties. I nearly always go for Religious Victories, so I think I would have noticed.