Try
sudo find /home/public -name "*.mp4" -exec mv '{}' /<dir> \;
Change the <dir> in bold to where you want the files to end up. Also, be aware that by default Find will not follow symlinks so this will only work on content local to the core, you need to add another option to allow it to follow links out to your network storage.
Finally, I would strongly encourage you to make /home/public more specific. Not sure where you are moving from and to, but you need to ensure that there is no chance that find will search/recurse into the destination otherwise it will find files again that it has already moved and try to move them again to the same spot!
Oh, and for tidiness, you really should go back with the web admin at a later point and clean up the metadata...