A SETTING WITH INTERESTING GAMEPLAY IMPLICATIONS:
Multiple parallel rings rotating at slightly different speeds, thus the neighboring region is continuously changing. In particular I think of a strategy game where trade and conquest happen with static neighbors across small borders limited by the narrow band of the ring and also with shifting neighbors that come into and subsequently leave range (the shared border growing and then shrinking).

Imagine you are trying to expand your continuous territory around your ring, fighting a neighbor when a more powerful nation hostile to you passes by, so you must divert to protecting that border while they are in range. Or an ally passes by and you get their support for a short while and can maybe make a big push. Or a region rich in a resource passes by and you trade what you can to stock up on it.