I would switch between scenarios. (every significant milestone, or around three or four sessions, but mainly whenever the current "dungeon", "book", or "adventure" is done.) I'd suggest taking a look at the books in the series- there are one or two long term issues throughout the books, but by and large each one is a mystery within the book.
What you need to make sure of is that you finish each adventure before handing things over. The villain might not be dead, but he's beaten for now. The city might not be safe, but this particular threat won't be what kills it. (Note: If the bad guys won, which they do sometimes, then that's an ending too.) In the section on cities, it points out that city aspects are expected to change as the game progresses. Basically, whenever one of those aspects changes, and everyone involved in the hubbub has conceded that there's not much more they want to try, then you can safely change. There are a few possibly hiccups.
First, what was the odd character out doing? In Dresden, you all didn't just meet at a bar, so when bad stuff went down and you all pitched together, why did one of you not show up? (That would be the DMs character.) This isn't a hard problem, but you do need to think about it. Did the bad guys start out by neutralizing them? Are they busy dealing with another part of the problem? Are they recovering from a consequence from last scenario? Or, are they right there besides everyone else? (Some people can play a character and DM the game at the same time. It doesn't work very well for me, but if you can pull it off go for it.)
Second, are there any secrets? Is one the the players related to one of the bad guys? Is the big bad working to find the three pieces of the triforce, one each scenario? Are the bad guys secretly breaking the seventh law? Is one of the players secretly a red court infected? (That one actually came up- A player got infected during a session, and hid it effectively for two scenarios. Then we switched DMs when the DM went for a semester abroad, and I had to deal with this sudden new plot thread, and pretending I didn't know when the DM got back. If you're rotating DMs, everyone just has to be okay with the fact that inside of three or four rotations, everyone is going to know whatever deep dark secret they were hiding.)
The best way to fix this is to just not have secrets, but that takes some of the fun out of it. Depending on the players, they could roleplay not knowing. (I love doing that sometimes- the example in the book goes something like "A ghoul snuck in the back window. You failed the alertness roll, so why didn't you notice?") Thing is, just because you're not letting it get transferred to your character, doesn't mean that it's as fun. I love surprises and plot twists, and that is something that gets harder to do as more people share the DM seat. One way to still have this is to have a bunch of bad guys, each with their own separate plots, each run by a different DM. Then again, maybe the big reveal isn't as important.
Another way, if you're all clever at this sort of thing and everyone is cool with it, is to retcon a little when you need. Anything that happened, happened, but you might try something like "Okay, so the first time it turned out Jim Blackhat was a sorcerer. Then next time in turned out he was also a redcourt, but he had been trying to keep it a secret! But now, in his secret lair, you find the true source of his power- He's been breaking the seventh law!" Basically, add your own twist every time the DM seat changes hands. Just be careful this doesn't get too tangled and confusing, and remember not to change anything that actually verifiable happened. Blackhat fell off a cliff, but survived and is back for revenge is probably fine, we shot Blackhat in the chest at point blank but he was a vampire and regenerated might be, but around the time the players burn the body and douse the unmarked body in holy water, he does not come back. That's why evil villains have apprentices anyway!
Third, all characters get access to all milestones, even if they slept through the whole thing. That sounds a little obvious to some, and wrong and unfair to others, but especially if there are a couple of significant milestones or even one major milestone, you need to give it to all characters. Playing more than a few skillpoints behind, or even one refresh behind, is an irritating shift in power and agency. Your mileage may vary on that, but it's not hard to come up with a story to justify it if you need.
In the end, it works out pretty well. Just realize that you are giving up a certain level of genuine surprise and twist to the story, and the rotating cast will feel a little different than the same tight nit team every time.