If you have two email address and one is gmail. it is best to set up email forwarding to gmail email address in the webmail account. Just send the emails from one server to another.

There is little point downloading to Thunderbird, all the emails from account 'A', just to forward them to a gmail email address and then download them again.

You can set up more than one mail account in Thunderbird and get both email addresses to download to Thunderbird, so you see all accounts in the same Folder Pane.

re :migrate Thunderbird e-mails

I'm presuming you mean you have another mail account (not gmail) which has downloaded emails to an exisitng mail account in Thunderbird.

If some emails are already downloaded to Thunderbird, you could create the gmail account and copy emails to the gmail account, but this would mean you are putting those emails onto a different server, so this might change all the dates on the emails. right click on email and use the 'Copy to' option.

when you are certain that they have been placed on the gmail server, then you could delete the original if required.