Interaction Gmail and Mail.app - Moving messages locally
you need to login to gmail's settings into the Forwarding and POP/IMAP and make sure you have it selected to to delete Gmail's copy, so it doesn't sit in Gmail's folder/server while you have it locally stored on your Mail.app, e.g.; picture below of option you need to select.

notice option &quot;delete Gmail's copy&quot;
once you do this you should good to go, hope this helps! :)