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") and you'll see the file there.