If your e-mail provider supports the IMAP protocol, you can sync your e-mails on the provider's mail server with a desktop e-mail client like Thunderbird. Once you have a local copy of your messages in Thunderbird, you can search them locally, back them up (copy the local files/directories), etc.

If you want to retain messages on the desktop and delete them from the mail server, you can move them into a local e-mail folder that's not synced with the server.

This is how I manage my e-mail (my e-mail provider is FastMail).