As far as Thunderbird is concerned, you can select and save the messages you want as single eml files. You can choose the Google Drive local folder as the target location for those files and they will sync to the cloud. You will of course want to organize the saved eml files into appropriate folders. This is a manual process and you'll have to manually select and save each new message that arrives into the appropriate folders. Alternatively, you could add a Gmail IMAP account to Thunderbird and upload (copy) your locally saved messages to appropriate folders of the Gmail account. Gmail's storage is facilitated by Google Drive, so essentially, your uploaded messages will be in the cloud for as long as your Google account is active. You can automate copying of newly arrived messages by configuring a filter to forward received messages to the Gmail account. You could also setup forwarding on the server-side instead of relying on Thunderbird (a client). Lastly, you can connect your email account to Gmail and have Gmail import your existing messages from your mailbox and fetch a copy of newly arrived messages automatically. This is automated in Gmail's server, which essentially turns Gmail into a client (like Thunderbird) through which you access your mailbox.