I'm using FolderSync for a similar purpose.
With that App, I can schedule the synchronisation of selected folders from my phone's storage to (in my case) a shared folder in my home network. The same is set up for my Nexus 4. By that, the photo albums on both devices are synced. I guess it should provide the options to sync also your Dropbox files. Give it a try!
Nokia 3210 -> Nokias 6210 -> Nokia 5510 -> SE W800i -> SE K850i -> HTC Magic -> HTC Desire -> LG Nexus 4 (Stock 4.4 + Franco)
Nexus 7 2013 (Stock 4.4)