Continuing the discussion from Apple Mail doesn't work with archiving mail:
Mine doesn’t do this, I have a fresh install now too, it didn’t do it on v.0.12, and not on v.13, and not on v.13a with my apple mail app — if I have a new item, and it unread, I click on archive, it will send it to archive folder and mark it as read instantly (no delays) - same if I were to delete it. I also tried this on the web site (round cube) and the results were the same, and (almost) instantly replicated on the mail.app client on the desktop. Maybe it has something to do with your imap mapping settings somewhere, but I think we checked those already.