Official comment
Hey Freester,
You can move the database to an external drive, but the best thing to when moving computers is to File > Library > Backup on the old, then File > Library > Restore on the new. Even if the music is mapped to a different letter, it can be relocated easily, but the drive MUST be mapped as a local drive, rekordbox doesn't play nice with network paths.
If you wanted to share the database, you could put it external, but I would recommend that you ensure everything else is the same between computers (such as the user paths).
An alternative is if you're using rekordbox 6, you could use the cloud sync function and avoid any headaches.