I think this would be a good time to implement clocks. The factions all have their own goals and agendas, now you need to determine whether they advance them or not. Maybe every day or something they have a chance at advancing their clocks. Let them all have a chance at the same time, but the odds can be different of course. More powerful or better organized factions, or factions with smaller goals might have better odds (4, 5, 6 on a D6) or perhaps factions dealing with internal strife and turmoil, or factions with loftier goals would have lower odds (5,6 or even just 6 on a D6).
Then I'd let the PCs influence these chances. Like if the PCs are actively assisting one faction they get advantage, or if the PCs are working against a faction they get dis-advantage.
You could even do fun things like if Faction A and B share a territory border, and A advances their clock, but B doesn't, A has a chance to expand their border.
There of course will always be exceptions (Like for example Faction A's goal is to obtain the holy relic in level 4 of the dungeon, and you've decided that filling their first clock will result in them placing a foothold in level 4, and the second clock is obtaining the relic. But then the PCs find and take the relic and just give it to Faction A.. There's obviously no need to roll to advance the clock.) but I think this helps it feel alive, mostly for the factions that may not be in the spot light. So if the group meets Faction C early on, but then spends a week in the dungeon before coming across them again, it would be neat to learn that Faction C maybe obtained a powerful weapon, or perhaps they were able to steal a large chunk of territory from a bordering faction so now they control a larger piece of the dungeon.
Honestly a big campaign set entirely in a mega dungeon controlled by rival factions sounds like a lot of fun.