If your goal is solely to transfer your email, your best bet would be to create a dummy gmail account with IMAP and add it to tbird as a separate account (so that it doesn't go to the general inbox). Once you had that done, in tbird copy all the folders you want transferred to the dummy account and wait for it to sync with gmail's IMAP server. Finally, add the dummy gmail account to your choice of email client on your phone and wait for the IMAP sync to complete.
If you want to continue to receive email on your phone from all these accounts, you can add them in as you like to the email client of your choice. Personally I'd recommend K9 for multiple email account management myself; it has an unified inbox similar to tbird's general inbox and is all around mucho handy.