Infantry allows cavalry to attack penetrated cities

I like the idea, but I'm wondering of an implementation where it's fair and still makes walls meaningful.  I'm imagining that an infantry unit standing inside a walled tile gains a status at the end of its turn "Opening Gates", and if it's not attacked while it has that status, the walls in that tile are considered breached after 1-2 complete turns?

TBH, I'd like a full rework of walls because by late game, it just becomes meaningless.

Walls should be a buildable that you build one per tile - at your discretion starting with the city center. And most people should just build walls around the city center, maybe extending outwards 2 tiles at most. The whole city shouldn't be surrounded by a wall. This would partially help the cavalry problem as units couldn't hide so deep in the city as currently.