What do you mean? By default, both Thunderbird and Mail.app should fully download and locally save an archive of your mail.
To verify this in Thunderbird, check out Tools -> Account Settings, then go to the "Synchronization & Storage" page. Make sure it's set to "Keep messages for this accounts on this computer," "Synchronize all messages locally regardless of age," and "Don't delete any messages." You can click the "Advanced..." button if you only want to sync certain folders.
To force a sync, you can go to File -> Offline -> Download/Sync Now, but you shouldn't ever have to do that with the above settings.
To verify this in Thunderbird, check out Tools -> Account Settings, then go to the "Synchronization & Storage" page. Make sure it's set to "Keep messages for this accounts on this computer," "Synchronize all messages locally regardless of age," and "Don't delete any messages." You can click the "Advanced..." button if you only want to sync certain folders.
To force a sync, you can go to File -> Offline -> Download/Sync Now, but you shouldn't ever have to do that with the above settings.