Interaction Gmail and Mail.app - Moving messages locally

I use the Mail.app to access my multiple email accounts on my MBP and store some of the messages locally so that it doesn't end up stored forever online. The thing is, I feel like moving a message from my Inbox to a local folder (i.e. "On my Mac") works for all accounts except Gmail. Whenever I move a message from my Inbox to a local folder (either by drag-and-drop or with a rule in Mail.app), I discovered that the email is still in Gmail. It just isn't in the Inbox anymore, I have to go to "All Mail" to see it. Is there a way to erase the message from the server as soon as it is stored locally? (or even with some delay like with deleted messages) Otherwise, it kind of defeats the purpose of all this... *I discovered that more than 1,000 emails are still stored online while I stored them all on my mac...*