How to backup live photo as 6s feature in dropbox or other
The iOS app Photo Sync can auto-upload Live Photos to your Dropbox Account and as an option delete those after successful upload.

The live photo will be – as usual with all these work-around solutions – split into two parts, a „YYYY-MM-DD hh.mm.ss.jpg“ and a „YYYY-MM-DD hh.mm.ss.mov“. (The naming scheme can be freely adjusted to your needs.)

Another seamless option is Google Photos. Works absolutely clean and straightforward. Live Photos can be viewed and played in Google Photos as used to from the Apple Photos app.