Syncing a git working tree with Dropbox?

I've heard of people keeping a bare repo in their Dropbox folder (https://www.dropbox.com/) and pushing to and pulling from it, letting Dropbox sync the bare repo between their machines. In other words using Dropbox as a form of hosting for a private git repo. What I want to do is sort of the other way around.

I keep on getting into the following mess: I have some changes in my working tree on machine A, I stop working on machine A and don't commit and push the changes (to my remote, 'central' bare repo) either because they're not ready yet, or I forget to commit or push. Later on I arrive at machine B which has its own clone of the same repo, but because I didn't commit and push the changes on machine A I don't have access to them on machine B and I can't continue working on them. The two machines are physically located far away from each other and they're not accessible over the internet. Argh!

Dropbox is a proprietary sync service that gets around this problem because it automatically syncs your files whenever you save them. But I still want to keep my project in a git repo. I'm assuming that keeping the actual .git folder in a Dropbox folder, so that when git makes changes inside the .git folder Drobox syncs them, would be a bad idea. It seems like taking two different synchronisation systems and mashing them into each other. But what about just the working tree?

My idea is that I keep my .git folder safely outside of my Dropbox folder, but my git repository has a detached working tree that is located in the Dropbox folder. On machine B it would be the same setup. So the two machines each have their own clone of the git repo and these are synchronised by git push and git pull with a 'central' remote repo. But the two clones share the same working tree, or more accurately their working trees are synced by Dropbox.

The working tree is just files, I don't see how it's different from Dropbox syncing any other files. Dropbox and git ought not to collide in any way. So this should work fine shouldn't it?

This way I don't need to commit and push my changes until they're ready/I remember to, but whenever I move from machine A to machine B my uncommitted changes will still be available to me because Dropbox has synced my working tree automatically.