Dropbox has an API that will (theoretically) let you do this, but there hasn't been a ton of people jumping up and implementing it yet. It'll be cool when it shows up.
In all cases you get a delete (if the file isn't new) and then a new file event.
I implemented it, and it kinda sucks for this sort of thing. The purpose seems to maintain local state to mirror the state on Dropbox. Not terribly interested in that...I just want to subscribe to specific events (webhooks, anyone?).
This was their solution to the frequently requested webhooks. It falls short. Way short.
I haven't tried it, so I don't know how gracefully it handles renames or moves.