Deleting messages and/or moving messages into folders does not carry over to my iPhone & iPad resulting in duplicate work - didn't used to be an issue - and YES - accounts are IMAP
iMac Pro
You will have to verify you are signed in with the same AppleID to iCloud with the same contact info.
>System Preferences>AppleID (if Catalina)
>System Preferences>iCloud (if Mojave or before)
Once verified I would simply sign out of iCloud on each device and sign back in again.
What to do after you change your Apple ID or password ...
Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV ..
Manage your devices in Settings on iCloud.com - Apple Support
That didn’t do anything except having to redo my wallet etc.
Deleting messages on my IMAC does not delete them on my iPhone or iPad and vice versa
