Did not find a lot on the internet on this particular error, so I guess it does not happen very often, but in my case it delayed to move of the mailbox in question by a week or so until I could resolve it.
When a mailbox is moving to a different Exchange organization (cross-forest or to/from Exchange Online) the move process copies the mailbox data to the target database and then right at the end of the move updates Active Directory in both the source and target forest. In the source it changes the object type from mailbox to mailuser (or remotemailbox if Exchange Online is in play, though this is really a special form of mailuser) and in the target, updates the mailuser to become a mailbox.
This particular error occurs at this stage. The Get-MoveRequest cmdlet reports Failed as the status, and Get-MoveRequestStatistics reports FailedOther as the status. If you get the move logs (Get-MoveRequestStatistics <name> -IncludeReport | FL | Out-File <filename.txt>) then in the logs you will see CannotEnterFinalizationTransientException as the error repeated many times until the move fails.
The fix for this issue is as follows:
1. Check that the Exchange System account has permission to the Active Directory object in question. In Active Directory Users and Computers choose View > Advanced to enable the Security tab and then view the security tab on the object in question. Edit > Advanced and then check or click “Enable Inheritance” option or button (depending upon version of AD tools). If inheritance is already set to enabled there is probably no harm in disabling inheritance, copying permissions and then enabling inheritance again.
2. Move the mailbox to a different database in the source Exchange Organization (New-MoveRequest <name>) and waiting for that to complete.
3. Removing and restarting the move in the target forest. If you do not remove and restart the move in the target you will see both MailboxIsNotInExpectedMDBPermanentException and SourceMailboxAlreadyBeingMovedTransientException. The first of these is because the mailbox is not where the target move expects it to be, and the second of these is becuase the source is currently being moved and so cannot be moved to the correct target forest at the same time.
This should resolve your ultimate move request – it did for me!