I have been promoting my own religion in my cities, but some have fallen to others' religions instead. I know that I can only buy faith-based assets in cities that are following my religion, and that my religion has certain benefits, but is there any other detriment to my city following another's religion?
Each religion has founder benefits, and follower benefits. Any of your cities that have a majority religion get the follower benefits of that religion, even if you didn't found it. This includes any buildings that religion provides, and any religious units you purchase in that city are based on that majority religion.
Any city that follows the religion you founded (whether it is yours, your opponent's, or a city state) grant you the founder bonus for your religion. The same is true in reverse - any of your cities that have a majority religion founded by an opponent, they get the founder bonus for your city as well.
The main reason that you would not want to have an opponent's religion in your city is to prevent them from getting the founder bonus. The main reason that you would want an opponent's religion in your city is to get the follower bonus for that religion.
IIRC you can see all of the bonuses provided by a religion in the religion window (accessible by clicking the faith meter at the top of the screen).
The Civilization Wiki lists all of the potential follower benefits. Here are some that would most effected by your city following an opponent's religion:
- +2 gold per city following this religion
- +2 faith for each foreign city following this religion
- +1 happiness for every city (every 2 cities in Brave New World) following this religion