Here's my idea:

I think influence should determine borders, but once influences tough- it should become the border. 

Rivers and other natural boundaries should be harder to spread influence into, so that those areas become natural borders more often.

City influences should determine provinces, which determines the territory gained/lost when a city is captured.  As with country borders, cities have individual influence, which ends when the borders touch.

Then again, I could see this tactic leading to ICS potentially, so care would have to be taken in regards to that.

While I agree with you, it sounds like an absolutely terrible game mechanic. I mean really, really dull. I don't want to sit at a table exchanging favors for the majority of the game.

Cultural land-gain does happen in the real-world, at least that is how I explain the EU to myself. Yes, some paper signing was involved too - but essentially it is everybody saying "hey, we like each other enough to jointly share mostly much of everything".

Yes, but that was political. It wasn't a matter of France building a museum in Paris and suddenly having their border expand to take over Belgium. That is how most games have done culture and borders, and it is stupid.

I would be ok with 'culture' spreading borders into frontier lands, but not into the territory of other players. That should only be accomplished through military or political means.

I like the idea.  I like anything that makes diplomacy more interesting.

I've always thought it a horrible game mechanic that cultural increases push borders back and forth with no consequences* in most games that feature the mechanic.  How many wars have been fought over where a border lies?  Once to borders meet, they should stay where they are until something actively changes them.  War, treaties, founding of another city, destruction of a border-producing city by a third party or monsters. 

This idea opens all sorts of more interesting diplomacy scenarios that coincide with much of how humanity has developed.

Edit:  *By no consequence I mean that neither side reacts instantly.  Border movement should be a BFD.  

Borders are more likely to form on natural features, so the influence cost to cross a border such as a river or mountain should be higher, to increase the likelihood that borders meet there.

I think this might be a neat mechanic for multiplayer but I don't see how the AI could realistically use it. Similar to the AI's inability to use teleport in any fashion, either offensively or defensively.

 Yes, but generally it's the people who live on the land that actually do stuff rather than the land itself. Cultural boundaries represent the influence of a government, which generally extends beyond it's national border. Doesn't matter where you arbitrarily draw the line on the map, if the guys on "your" side of the line are more likely to listen to the government on the other than your own. It's probably not as apparent in the Americas due to it's history, but if you look at the border regions of Europe for example you can see it in effect (particularly when you get down to regional levels).

 It's problematic because games tend to roll up the concept of influence, culture and borders into one, but any way you implement it you can argue is just as silly. Generally speaking borders tend not to be quite so solid as they look on a map, particularly for the population in the area. It really depends on what the gameplay effect of the border is; given the subject matter the approach of Age of Wonders might make the most sense - your border stops at the point your sovereign's spell casting range does. After all, a mountain range or river isn't much of a boundary if you can instantly teleport troops over it.

The spreading of culture should not change political borders. And culture affect would vary based on how open - or - closed borders were between the two 'nations/kingdoms. 

Generally, a strong culture, exerting its presence beyond a political border should have effects on both the owner of the stronger culture, and the owner of the weaker culture.  It should affect income from trade routes, (higher gets more).  Ease of spying higher is easier to spy, lower is harder.  Also, perhaps a stronger  culture level will allow 'free' info gathering across the border.  Also, if the border is not closed, stronger culture will attract population from weaker culture to move to stronger culture cities (say, in civ, actually remove pop from cities and place it in stronger-culture  civ's cities).

The old border/culture dilemma.  If I may shamelessly promote a former but original solution: https://forums.elementalgame.com/388453

This is a must have for 4x games!  Why hasn't it been done already? GAME EVOLUTION DEMANDS IT!

@rogue captain  great idea s in link.   Yes.  Another aspect of this might be the 'limited/regional war.'  The idea here is that a dispute over a specific area is happening.  Both (all) parties agree to limit the conflict to that area.  Total war actions allowed in that area, but no where else.  How to define 'area?'  Perhaps use the city in question as a source of borders emmenating for it, separately from the 'sides' involved.  Perhaps a mechanism could be set up to make, for the duration of the limited war, a change in status of the city and its area.  The city would become its own 'nation,' with its own 'influence/borders.'  However it would be totally allied to original owner, so original owner gets all benefits of friendly zone of influence (dying sov gets resurrect, etc.) and enemies get negatives of being in the zone. 

Just an idea...

yeah i came up with this same idea for Civ 4 about 5 odd years ago on civfanatics, but if i remember was shot down for it... i thought, and still think its an obvious 4X mechanic...