Actually tested this myself.
Test 1
Started putty in windows
Logged in to Tower
Typed MC
Started copying file from one mounted unassigned disk to another one
Closed putty
Copying stopped - clearly depends on solid connection between machine with putty and UNRAID machine.
Sounds like a receipt for disaster if moving large amount of files.
Test 2
Logged in to UNRAID machine through web browser
Opened Terminal
Typed MC
Started copying file from one mounted unassigned disk to another one
Closed terminal window
logged out from UNRAID
Success - Files are copying in the background, about the same result as "nohup mv". Just no idea how to see the progress, since going back to MC in terminal doesn't show anything.
nohup mv /mnt/disk1/Movies/* /mnt/disk2/Movies &