The standard mail client will pull mail from the server for every “x” days. If you delete them on the phone, but it is not set to delete off the server, then they will re-download until they are deleted off the server, or until the “x” days expires.
I use e-mail on the phone as a canary in a coal mine so that I know what’s happening at the office before I get there, and to send quick replies. I only have the client pull 1 day’s mail, so at midnight my mailbox is clean.
I know that’s not the way most people use e-mail, but I hope it helps explain why they keep downloading.