These were part of a challenge--Remix Redux, which used to happen every year, and then the mods got burnt out last year, but some people revived it this year. It is multifandom and has quite an elaborate set of rules (profile here)

There are single-fandom ones, too--I think the Sherlock fandom runs a big one.

I find it enjoy it. Partly b/c I suck at thinking up my own plots/scenarios, and like working in the stricter confines. I've written a few stories I quite like as remixes.

You have to not mind people messing with your own stories, though--which can be weird, and also fascinating.