If the database is not too large and will never be updated on multiple devices at once (which will require a more complicated synchronization) then you just need to copy the database file from one device to another.
Dropbox Sync is a good solution for this requirement however this API will be removed by Dropbox next year so I don't recommend using it for new projects.
A simple solution that will require no backend at all is to let the user send the database by email.