the problem I noticed is Big Sur requires APFS on any new SoftRAID volume, but allows you to move a pre-existing time Machine from Catalina to Big Sur. I am speculating somthing related to this issue is happening to you.

# tmutil inheritBackup /Volumes/Time-Machine-NAS/OldMachine.backupbundle    
Mounting disk image...
Enter password to access "OldMachine.backupbundle": 
Failed to inherit machine store at '(null)', error: Error Domain=TMStructureErrorDomain Code=7 "Structure has wrong type" UserInfo={NSLocalizedDescription=Structure has wrong type}
#

But why was the old machine running Big Sur as well able to continue using its HFS+ formatted TM volumes? It can continue using it on Big Sur upgrades but not move it to a new machine using Big Sur? If true, that's pretty bad, Apple. (pun intended).

Yes, this is an old MacOS Extended (Journaled) volume, not APFS. But it was working fine with Time Machine on the old iMac (also running BigSur 11.4).

What's really frustrating, is that I was letting Time Machine alternate between this disk and an SMB volume on a Synology NAS, and Migration Assistant can't use that backup either. It sees the NAS, lets me authenticate to it, select the TM volume and then just sits there forever saying "Loading backups..."  I've seen that exact problem mentioned by other users on Synology's forums, and I have a tech support case open with them waiting on response.

But this is maddening that I can't get Migration Assistant on the new machine to adopt either the local TM volume from the SoftRAID disk or the remote sparse bundle one from the NAS. I periodically used both of these backups to restore specific files sporadically on the old machine -- and I watched the logs, so I'm as certain as I can be that both backups are in good shape. I really think this is some kind of Migration Assistant bug for the local volume (maybe you're right about MacOS Extended vs APFS) and a Synology bug for the remote volume. Aaargh.

I tried doing an end run around the Migration assistant by using 'tmutil inheritBackup' directly from a root shell but got a weird error (which I also saw others getting on Google with no solution) -- which definitely lends weight to your MacOS Extended vs APFS theory.

# tmutil inheritBackup /Volumes/SoftRAIDVolume/Backups.backupdb/OldMachine 
Failed to inherit machine store at '(null)', error: Error Domain=TMStructureErrorDomain Code=7 "Structure has wrong type" UserInfo={NSLocalizedDescription=Structure has wrong type}
#

If you run Disk Utility, First Aid? does it fix anything?

Is this an APFS volume? This could be a bug from legacy volumes, as Big Sur requires all new Time Machine volumes to be APFS.

It worked, but I only did one backup. I will try tonight after working on the system, so I have an incremental worth testing.

Yes, Apple had me run Disk Utility's First Aid on the TM volume. It found no errors and changed nothing.

I need to run the Migration Assistant directly after booting into an existing system in order to access the latest incremental backup. This should work but does not. Are you saying it did for you?

the first step was booting into internet recovery mode. Seeing if a user can select it there. You may want to try it, to be sure macOS sees it as OK. I was checking where the bug was, assuming this was a generic bug.

When I ran Migration Assistant, as you, the volume appeared and allowed migration.

What if you run Disk First Aid on this volume? Does it give any errors? (Disk Utility)

Boot into Time Machine, to be told you must use Migration Assistant (normal behavior)

What do you mean precisely by "boot into Time Machine?"

Here is precisely what is not working for me:

1) Time Machine backup on SoftRAID (RAID 5) volume on machine 1 (Late 2014 iMac), attached via Thunderbolt 2. Running Big Sur 11.4.

2) Reattach volume via Apple Thunderbolt 2 -> Thunderbolt 3 adapter to machine 2 (Late 2019 MacBook Pro). Also running Big Sur 11.4.

3) Machine 2 already has a running system. SoftRaid volume is visible in Finder and appears fine (has Time Machine icon).

4) Run Migration Assistant directly on Machine 2 -- it closes all applications and logs out.

5) The volume never appears on the list -- just says "Looking for sources..." forever.