Preserving years-long message thread from iMac to iPhone using iMazing

I have a years-long message thread [communicating with another person who uses an iPhone] that is still viewable on my iMac, but is reduced to just the last three message exchanges on my iPhone.

At this point it matters less to me _how_ this happened and more important to save the message thread on my iMac, transferring it to a separate location for archiving.

I know Apple doesn't make this easy -- is this a task iMazing could handle, or is there some other method?

Rhet

[Re-Titled by Moderator]

Original Title: Long message thread is visible on iMac, but lost on iPhone. How do I preserve iMac version?