OS X Screencast to animated GIF
gist.github.com
gist.github.com
However, I can see why this could be a cool learning experiment.
If you're worried about size, just use the mp4.
I personally find it more useful than this OP project, and so wanted to share in case others also find it useful.
Works super great!
Anyway I bet an actual live broadcast wouldn't be too hard to do, although your poor viewers' browser caches would fill up pretty easy. But then that's what multipart/x-mixed-replace is for.