At one point, I designed a “procedure” for a “rotating GM campaign” for the the game Red Box Hack. Unfortunately, it was housed on a wiki which, as far as I know, is entirely lost.
The concept, though, was that the group of adventurers would have other missions and concerns. At the beginning of each “adventure”, they’d pick one member of the group to go away to accomplish something important (deliver the letter to the Princess or whatever). That character’s player would then become the GM. (The game had a series of prompts for determining the adventure that were structured as a conversation/debate between the adventurers. You might say, “I think we need to go to the Dark Forest,” and I might answer, “I don’t know, my friend, I heard there are giant spiders living there!”, and so the adventure would take form.)
I never got to playtest it, though.
I’ve seen one consistent “rotating GM” campaign, and that was the Grey Sands campaign, using B/X D&D. Many people in the group had adventures on the go, and a stable of characters. So any given person in the group could just say, “Hey, there is an expedition to the East. I’ll be running it at 9pm.” And then whoever wanted to play could join in. If their character(s) wasn’t/weren’t available, then they could still join in by rolling up a new one (and adding it to the character stable).
This meant that there were multiple storylines and adventures going on all the time. However, it was definitely facilitated by the structure of the game - multiple adventures and a character stable - rather than one, single, ongoing story.
There is a game called In a Wicked Age… which is intentionally structured this way. The first two sessions (Chapters) must be GMed by the same person, to set the tone, and then you’re free to rotate as you wish.
