Passage of Time
Time continues without pause while you are in a dungeon. Ideally you want to clear dungeons as quickly as possible so your followers don’t begin to starve.
Cult simulation continues as you are on your crusade fighting or not. So eventually if going slow you will start receiving notifications that cultists are starving, elders are dying, shelters are breaking and so-on.
That said if you leave your cult with topped off faith and hunger, you should have enough time for 2-3 dungeon runs (depending which, 3 being the maximum amount you run in succession anyway). Or if feeling optimal in preparation: cook food with no chance at illness for the next day before you head off.
Doing the brainwash ritual, as well as the fast ritual ensures faith and hunger never drop, which can extend the length of a run by a lot. Nothing can be done about Elders dying, I don’t think, besides killing them off before you go on a run.