Previously had similar activity in Mac mail, but have all sounds off, so cannot say it is exactly the same.
Result of combination of using multiple devices (iPads, iPods, computers), multiple users sharing a mail account, and using a pop3 based mail account (yahoo). Pieces of mail get moved between folders on the web server when using other devices, then when you open Mail on Mac, you see a bunch of activity as it sorts the mail. When using Mail for a pop3 account, you see all the folders that exist on the web page down the left sidebar of mail. When you move mail to trash on other device, you will see activity on Mail next time you open it.
All activity (and visibility to online folders) stopped when I changed to imap.