Multiple path progression question - Fortune's Weave

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I'm not sure if this has been answered before but I do have a question when it comes to progression. I started off with Cai and I'm enjoying his story line very much! I'm a sucker for the whole kind island protagonist but with a dark past.

Let's say I finish Cai's path and I decide to start Theodora's, what exactly carries over? Does the game do anything to help mitigate the repetitive aspects? What happens to dungeons you've already explored, for example?