Too often by mid/late game I find my culture and religion has spread over the world without little to medium effort on my part.  By the same token, I'm sometimes seriously challenged by being snowballed when I don't pay attention to culture or religion, because all it takes is a toehold for things to get bad quickly.  The reason for this, primarily, is that once a territory "converts" so that it's majority is held by another culture/religion, it stops lending its support in influencing neighbouring territories.  Once one or two key territories are "shut down", the neighbours seem to fall faster and faster.

What if territories continued to spread their culture to adjoining (and trading) territories regardless of the dominant culture?  Would that help protect every empire from being snowballed whoever takes the first lead in the culture war?

The Dutch have only one unconverted territory left, and it's getting 0 help from its neighbours.

Once they start to fall, the snowballing becomes bigger and bigger.  Each territory is left with its own influence (sometimes quite respectable!) standing individually against the influence of all 28 territories of my empire.  They may not all generate a lot, but even 2-3 cities vs. 1 city will eventually completely lock out an empire's ability to share influence among its own territories.

What's been everyone else's experience with this, and if you've experienced snowballs (from either side) how have you tried to manipulate it?

I haven't thought about this much till now... but when I take cities, I usually win by routing the enemy completely instead of holding the flag, and this includes eliminating the population turned militia at the time. This means newly captured cities start at 0 pop and emit much less Influence, so I'm assuming that makes it easier for other cities (such as my standing cities) to spread their culture.

This is my experience too. I don't know whether to call it good or bad.

But it does mean your strategy does need to at least focus on influence - otherwise AI will have alot of grievances against you and it will be impossible to play peacefully.

It works for me since I love having influence anyway for expansion. Religion is just another bonus if you want to play that.

I will say the OP has an interesting idea which may prevent snowballing and make this aspect of the game more challenging. Does anybody see any unexpected consequences of making this change?

With influence over territories being a part of WS (again, iirc, as I'm pretty sure that was the case in OpenDevs, because I recall saying it's interesting way to simulate local resistance movements), it would be good for foreign territories to be tougher to flip and more difficult to keep in your influence sphere. They shouldn't produce Influence for you at all if not owned or not IP, and shouldn't produce 'your' Faith unless owner is converted to your religion.

I still think that we could use an active part for both Influence and Religion, with being allowed to focus one territory (separately for Culture and Religion), boosting your power there, providing that AI can be taught to be smart about using this.