Ok, this is an interesting answer, but I think it doesn't exactly engage with what I'm saying. I know precisely what you're getting at with the above, but I feel like "waiting in a room for team PC to arrive" is just a matter of deftness of GM framing.
For instance:
I have a dungeon with goblins. It is generaically themed (eg its not "Cooking the Caraveners"); "Goblins in the ruin!" Its basically an unthemed, off-line dungeon that is only coming on-line right now because the PCs are interacting with it. I have it mapped, keyed, stocked and ready for play.
I have an opening situation in room 1. I have a situation in 5 of the 20 rooms (every 4 rooms) with the Wandering Monster Clock handling the dynamic content generation. Even if they heard about this dungeon multiple weeks prior Why would I need to change the opening situation in room 1, even if the PCs heard about the Goblin in the Ruins (!) dungeon x days/weeks ago, when the dungeon is only now coming on-line?
Rooms 2 through 20 may have dynamic interaction because the dungeon has come online and perhaps the results of stock encounters or Wandering Monsters changes the situation of subsequent rooms (this would be the Setting/Faction Clocks I posted upthread - these came on-line after PC interaction).
But why do I need to change the opening situation to the prior off-line, now on-line content? Why does the framing need to be evolved (and something like, "well because offscreen/off-line thing y or z occurred" is not responsive to the question...that is just a statement akin to "well, because Setting Solitaire"...which is fine...but then "why Setting Solitaire?").
I think I disagree with "is just a matter of deftness of GM framing." Unless I misunderstand your point. The point of living monsters, living NPCs, is they in effect have agency. I am not saying every single situation gets handled this way, because there are always exceptions, always things that you do out of convenience in play, but ideally, the notion is instead of prepping static situations, you are loading the game with NPCs and creatures who have clear enough goals on their own, they are something the the PCs interact with, not stumble into all set up and primed if that makes sense. Sometimes though, with dungeons, for convenience, I think this often happens less. You can do it in dungeons too, there are ways to set up a dungeon so it is more like a lived in residence than a place where the inhabitants wait in a room for you, or situations are held in stasis for the party, but sometimes it is just easier to say "there is a poisonous demon monkey in the grain barrel foraging for food" (and I have certainly done that from time to time)
but like I said earlier, you can evolve these things as much as you like. It isn't a law code you have to abide by. Everyone takes a slightly different approach. I think with a case like an initial situational, you may be fine not evolving it. You really have to judge that for yourself. But I do think, overall, if you are not evolving things, you are going to miss out on some of the life a living world can bring (for the reasons I stated in my previous post). And again, ideally the characters and factions are clear enough in motives and goals, that even if you are using some kind of table to trigger changes, it is pretty easy for you to logically figure out how these forces are playing out in the setting even if you don't rolll).
If I understand your situation correctly though, the players having heard about something three weeks ago, may in fact be a good reason for evolving the situation. I can't really say based on what you've said here, only you would know the answer really in the context of your campaign. But if I have a rumor that the players encounter, like General Lai is contending with a group of rebel bandits who a fortified in a trapped forested hill. When the players get there if he is still at the bottom of the hill and hasn't done anything in those three weeks, they may get the sense it doesn't seem like I considered anything that ought to have happened in the intervening time.
The answer to your question is the reason you would do it, is to prevent the players from feeling like the situation was held in stasis before they arrived like in a video game. If you don't feel that is going to be the case do what you want, if you feel the trade off in fun, isn't worth the effort and annoyance of evolving the situation, then just stick with the situation as you set it up. This really isn't meant to be some kind of straight jacket. It is a tool. Take Feast of Goblyns for example. There is a dungeon in that adventure with stuff keyed to it. But at least when it comes to the major NPCs, you are expected to move them around more so they have life and don't just feel like they are waiting around. For me, that is the starting point for this technique. Then it is just a matter of how far you want to take it. I would never tell you "every dungeon you make has to be evolving constantly". I think that would be rigid, bad advice. But I would say, if you want a living world, try evolving some of the situations if you can. If the players go somewhere and it strikes you there should have been developments to the situation they are heading in, that is a perfect time to try it. Or you can always structure you dungeons so they are more like homes (i.e. people aren't pinned to a specific location---instead maybe everyone is on some kind of wandering encounter table, or when the PCs arrive you roll to see who is there, who isn't; etc). Also if you have a conflict in a dungeon between factions or something, when you set it up, you can do it in a way that it is easy to evolve (i.e. by spring the Goblins take half the dungeon, by summer they take three quarters, etc----this can also be left to rolls: roll each season to see who gains more territory). Like I said, these are all tools. I think at the end of the day, if you are using a living world it shouldn't be done in a way you think harms play at the table. And sometimes you can mix and match structures. You can run a sandbox that has the occasional structured murder mystery in it. Sometimes when we talk about these things they come off as codes you must obey, and philosophies that cannot bend. I think that is a bad way to go. Sometimes you are going to find it easy to evolve situations and settings 90 percent of the time (for example when I am dealing with sect factions I find this very easy to do), other times the percentage could be 10%. One of the reasons I have tables for this sort of thing, is precisely so I don't have to constantly be checking in on and maintaining these things if my campaign is getting complicated. But definitely always try to keep those core NPCs active, generally prepare goals and intentions, rather than prepare events or set pieces, for a living sandbox.
Remember my main contention in response to setting solitaire was 'this isn't really solitaire'. I wasn't saying you had to evolve every situation 100% of the time 'because'. How you use this set of tools is going to be up to you if you try doing the living adventure thing. But even distant events and developments can be relevant to the party. If moving this stuff in the background was totally pointless and only for the GM, I would have stopped doing it a long time ago. It is because I've found it adds to he players' experience of the world I continue to do it.