Are you using POP3 or IMAP? Basically POP3 delivers the e-mail to your computer without storing it on a server, with IMAP the e-mail is on a server and you view it using your e-mail client. A more “in-depth” description can be found here if you are interested.
So I’d recommend using IMAP if available because then you can first read the e-mail in the sandboxed Thunderbird and then delete the unwanted e-mails (this will be reflected on the server) and then open Thunderbird unsandboxed and then that client will be able to read the e-mails that you did not delete. But there might be some things you don’t like with this method.
Otherwise if you only mean specific e-mails and you want to save them as files then you can go into Thunderbird, right-click the e-mails you want to save, then click “Save as” and then save them to the Shared Space, you should be able to open these in the unsandboxed Thunderbird.