The main problem you are facing, i.e. backup of mutable files, does not depend on docker. Use a tool like rsnapshot or dirvish to make backups in the volume, and then use the approach described above to move these backups somewhere safer like Amazon s3 or glacier, based on your reliability requirement.
If you mount volumes from another container or vm host using the -v switch, file changes are reflected in all containers (or vm host) more or less in real time. (There is some delay due to AUFS that docker uses over fs hosts, but not so much). If the backup container is running continuously, it can continue to make backups, and the files will always reflect the latest files seen by the mysql container.
Edit: for clarity.
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