"Live" Photos switched to JPEG on import from Messages to Photos on Big Sur

[ThrowerGB]

"Live" Photos taken on our iPhones (12 Pro & 11 Pro Max) and sent with Messages (iOS) are converted to JPEG on import to Photos in Big Sur. Thus losing the "Live" characteristic. The "Live" characteristic exists when viewing the same photos in Messages in Big Sur.
When the same photos are saved from Messages in iOS and then viewed in iOS Photos, the "Live" characteristic is still there. However, I don't know how to tell what format it is in iOS Photos.
Do I have a problem with settings or is this a bug/inconsistency of some sort?

 

[Slartibart]

just continue here...

 

[ThrowerGB]

just continue here...

Thanks for taking the trouble to respond. Unfortunately the thread you point to is addressing a different question to mine. But I understand this is complex. Apple hasn't made the various ways of transferring images clear and consistent. But perhaps I didn't make my original post sufficiently clear and specific.
My issue specifically addresses an issue with the transfer of images from Messages to Photos. There is inconsistency between the results on MacOS and iOS. The MacOS version doesn't maintain the "live" part of the image. The iOS version does.
Again - thanks