Is there a time limit for anything?

I've noticed that every time I enter a dungeon or have a battle, days pass. Since it takes a while for units to unlocks supports with each other, I've already done quite a bit of grinding in the first dungeon.

If I continue to grind out levels in a dungeon without advancing the story, is there a possibility I'll reach a game over scenario due to not doing something in time? Or is the calendar simply there for flavor?

Solution 1:

In act 3 onwards, each "day" effectively represents a turn and on each turn an enemy unit group has a chance to spawn at the acts final map and these spawned units can move to other maps, even cleared maps. They can be annoying to deal with if they join other groups of units but on there own they should be fine. If a Cantor spawns, they can be used as a handy, albeit longwinded method of grinding without using multiple days. Just position your units around the cantor and kill the terrors as they spawn. This probably works best if the terrors spawned are revenants as at this point in the game they are no threat, even if they don't give much exp.