I've used Outlook 2010 for years with a few gmail accounts. I have a backup .PST file on my local computer that when my email account got close to filling up, or I wanted to archive old emails, I'd move the folder/labeled emails from my gmail account to the local .PST file. When I did that, it looks like it would basically just copy the folder to my computer and then everything I "moved" from the gmail account it would put into the trashbin.

At work we've recently updated to Office365. When I move a folder, the same way I've previously done it, it would create a copy in my computer and just remove all the tags and the emails would be archived, the storage space wouldn't change.

Is there a setting I need to click to remove them from the server when I moved it to my local computer? I can't see to find one.