To move between Dropbox Folders on Android:
- Open Dropbox
- Tap source folder (e.g. "Camera uploads")
- Long-tap desired file
- Tap Move
- Tap Up to Dropbox
- Tap destination folder (e.g. "Public")
- Tap Move (screen returns to source folder so scroll to top)
- Select Up to Dropbox & destination folder (e.g. "Public")
You should see the file there.