You could give that a shot.  The "save folder" for flash stuff one of those assets folders in %appdata% somewhere.

If you're on Winblows, go grab yourself the simple symlinker program from here: https://github.com/amd989/Symlinker and make sure to run it as administrator.  You can do the same from an elevated CMD prompt, but why bother when there's a slick GUI.

Anyway, close your browsers, cut/paste the correct chat.kongregate.com (or whatever) folder from whichever assets folder you're using into your dropbox folder.  Run the symlinker tool and re-create your chat.kongregate.com folder as a symlink to your dropbox folder.  Like magic, your shit'll all write to your dropbox.  Rinse and repeat on other computers and you've got your SOL files syncing everywhere.