Now, this concept is very interesting to me. In a vague, hypothetical sense this sounds cool, but I can’t picture how this would actually look at the table. Would you be willing to elaborate on this, particularly in terms of specific examples in actual play? What action steps should a DM who wants to try running a game this way take?
I can speak to this. This is the style of campaign that I run.
So for me, the way it works is this: I have a ton of players, too many to cram into one group. So different permutations of players form several different groups. Let's call them Teams Alpha, Beta, and Delta.
Alpha contains Joe, Aaron, Emmett, Big Aaron, Pam, Chris, Sue, and Jeff.
Beta contains Big Aaron, Jeff, Shawn, Joe, Laura, Aaron, Emmett, and Joey.
Delta contains Pam, Aaron, Emmett, Chris, Sue, and River.
In all cases, the groups feature occasional crossovers and transplants. For instance, Jeff's pc Haji has adventured in Alpha, then moved over to Beta for a while. When he did so, Jeff's existing Beta character moved into Team Alpha.
Now, the Beta group is looser than the others and includes at least three groups of pcs, all of whom include Big A and Jeff. Aaron and Emmett have a couple of pcs that are in one of those groups; Joe and Laura are in a different one, but those two parties are about to merge. So when Beta plays, the actual group we run depends on who is available.
In practice, this means that no group gets as much gaming in as I'd like (other than Beta, which has the easiest and most manageable schedule collectively and is most able to come up to my in-the-middle-of-nowhere house, whereas we otherwise mostly game at Pam's actually-in-town place). However, there's a lot to love about this style of play. First of all, you don't usually have to say, "Sorry, there's no room for you at the table" when someone new wants to join- it's more a matter of finding which group has room for another players. Second, you can have arcs in the campaign where the pcs are affecting each other (but watch out for screwy timeline issues!). And finally, you have things where groups can see how other groups have impacted the world.
For instance, the first pc to hit 20th level (and semi-retire) in my game is a cleric of Jeff's. This guy now has a temple out of town and he uses divine intervention to constantly improve it. He has a crowd of peasants outside his temple every day, because he uses his tremendous holy powers to help the poor, heal the sick, etc. Other adventurers drop in when they need something from him, usually have to wait a day or so until he has some spell slots, and often donate or sell diamonds to him to prepare for the day when they need him to raise them from the dead. (There's a significant diamond shortage in the area after he bought up all the ones he could find for a true rez some time back.) So he has improved the city's overall health and the dynamics of its population dramatically.
The main advice I have for a DM who wants to run this style of game is to develop a setting with lots going on. If you have multiple groups of pcs, they need lots of adventures to go through- and you shouldn't mess around with running multiple pcs through the same adventure, you end up with messy contradictions and duplicates of treasure, different outcomes, etc. Let each group do its own thing. DON'T try to set them against each other unless you have the full buy in of the whole group. Rivalries are fine- but don't force the groups to be enemies. Better still is if they can sometimes team up and trade members.
One tool I recently started using that is extremely helpful is a campaign calendar. Print it out, write which group is where when on it. Then you know that, for instance, Team Alpha can't interact with Beta for another game week, because Beta has already played that week out.