In regards to:
What would be the best way to backup MySQL data before moving the mount point? Can this be done with MySQL running? Would a simple rsync off to another server work?
You can also backup your mysql database directory using rsync just add -a option in your command to preserve owner & permission.
rsync -arvz source destination.
Simple backup with MySQL downtime
Backup using XtraBackup
Percona XtraBackup can be used to take online hot (non-blocking) backups of your database.