Sounds like the essential structure for a "branching story", just done in Flash. Someone with Flash Generator experience should be able to set it up easily. I can also envision a way for it to be done with careful use of Perl CGIs and a strict naming convention using Flash's LoadMovie feature.
The "based on a snippet" aspect sounds analagous to the Exquisite Corpse technique of only seeing a sliver of the adjacent drawing.
I have been doing some Flash collaborations with a friend in New Zealand. The allure of doing a collaboration in Flash, to me, is the ability to share the library of symbols, to use other participants' symbols in creative or experimental ways. Maybe we could set up a collaboration where each contributor is allowed to create a limited number (3?) of graphic elements to put in the "stew". All contributors have to build new symbols and movie clips based on those simple/early elements.
Seems like it would better keep cohesiveness that way.
Let me know if you start a Flash collaboration? ...e