Forget Media tool you can’t move files with it just delete them.
Avids answer to moving files is to use consolidate but that’s pretty cr*p as a solution in your case.
I’d open the Avid media files folder on each drive. Then open the MXF folder you shoukd have a 1 folder and possibly some other numbered folders. Possi”bly even a bg_1 folder.
With Avid not running rename those folders by using a prefix number to present the project.
So maybe 100 is project A so you now have 1001 for the folder 1
And 1011 for bg_1 1002 for folder 2 if you have one.
For project B rename as 2001 2002 etc.
You can now create the Avid mediafiles folder and in it your MXF folder and then move those newly numbered folders to their new homes.
Remember that the next time you create media Avid will remake a new 1 folder and put the media in there. Any ba k ground renders end up in a new bg_1 folder.
Of you want to retain separation of media between projects when you finish in project A and exit Avid rename those new 1 folders following you chosen numbering scheme. So maybe 1003 or 1004 or even 10099 at some stage for project A.