I've noticed that Mail does the same thing on my computer and I searched around a bit looking for an answer. Most of the posts included problems with Mail's functionality, which I was not experiencing. It seems you are in the same boat, everything works, but that annoying little "downloading messages" dialogue pops up, every couple minutes it seemed in my case, without any actual new messages arriving.
Anyway, I found an old post from 2010 which offered a solution and since it was easy to implement I gave it a try and it seems to be working. Curious if you'll have a similar result and since I have no explanation of why this works (maybe) then perhaps someone else can still chime in.
Here's all I did:
- Go to my Junk folder and delete all contents (from all associated accounts).
- Go to my Drafts folder and delete all contents (from all associated accounts).
- Go to the Mailbox dropdown in the menu toolbar and choose Synchronize All Accounts.
The old post from 2010 claimed there was some problem with Mail and the Drafts and Junk folders...like I said it didn't offer much explanation. So far I have not noticed any more "downloading messages" alerts while in Mail and as I said they were coming up very frequently.
Hope this helps!