If the mail messages are being stored on the gmail mail servers and are accessable through the gmail web interface and the iPod Touch, and assuming you have gmail access set up in Apple Mail.app.
If that's the case, there's no migration here. More likely than not, the POP or IMAP SSL/TLS username or password is incorrect, or the port setting or the mail server host setting is not correct within Mail.app.
Use Mail's Window > Connection Doctor to check the settings. If Connection Doctor shows errors, open the details drawer, re-run the connection test, and then post the errors you've received after removing any personally-identifiable information.
The IMAP settings — for receiving messages from the mail server to the mail client — are here, and the mail server settings — for sending messages from the mail client to the mail server — are here. You need to have both the send path and the receive path — separately configured — correct, or your mail client will not receive messages or view messages already on the server, or your client will not be able to send mail.
The gmail sent folder setup can be a little wonky, too.
When given the choice while configuring a mail client to connect to a mail server, pick IMAP and not POP. Gmail does offer IMAP, too.