Here’s an AppleScript to sync IMAP mail folders (one way only, although it’s trivial to extend it to sync both ways):
tell application “Mail“
set destination to (every message in mailbox “Archive” in account “Home” whose deleted status = false)
set destinationIDs to {}
set d to a reference to destinationIDs
repeat with eachMessage in destination
set i to message id of eachMessage
copy i to the end of d
end repeat
set origin to (every message in mailbox “Archive” in account “News” whose deleted status = false)
repeat with eachMessage in origin
set i to message id of eachMessage
if d does not contain i then
duplicate eachMessage to (mailbox “Archive” in account “Home”)
end if
end repeat
end tell
(I use this to back up my newspipe archive to another IMAP folder for safekeeping, hence my being happy with it only syncing one way, but enterprising folk will love extending it into a full-blown IMAP sync tool..)