in a direct way you'll need something like FLV encoder solution from inside of flash application, but it will be too hard.
in a second way you may use some server instance (FlashMediaServer/Red5/Wowza/Stratus) to record events (read as Whiteboard Application)
So you will able to show to end-user his actions like an animation, not exactly as movie, but just an application with messages from your server, and their playback. Something like this.
in all other cases you will need GIF Encoder to produce some huge animation file, look google for it. http://google.com/?q=flex+GIF+encoder