There is no mechanism to push updates to your application. Only the desktop client gets async updates. It will be a glorious day when such an API exists.
But don't hold your breath, it's been a topic on their forums for years.
I'm a big fan of using dropbox like this, and am hoping to explore this approach in a lot of productivity apps i plan to make in the future. I like the idea that the user can 'see' their data in a readable format (json,csv,yaml - i still haven't decided which i prefer!) in their dropbox\apps\MyApp folder.
It's like icloud but you feel in control, rather than 'my data is out there somewhere but i can't see it'.