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Go to your providers Webmail site, log into your account there and check your mail. There are no Thunderbird Mozilla email. Emails come from your email provider. Thunderbird is email client software running on your computer.

re : I need to find out how I can access my Mozilla Thunderbird emails from an external/public computer.

You cannot access your Thunderbird program from another computer because you would need to leave computer running, logged in to User Account with Thunderbird running and set computer to never sleep etc and use a means of accessing that computer over the internet. Not exactly a secure method.

In Thunderbird, do you currently use IMAP or POP mail accounts? If IMAP mail accounts, folders containing emails are stored on the server, so you can access yout webmail account via a browser on any computer. If the computer is designated as yours then you would probably already have a 'User Account' setup. In this instance you could download Thunderbird if it is permitted and set up imap mail account to see same folders and emails. If it is not your computer -- a general use one, then suggest you access webmail account via a browser and make sure to clear history and cache after logging off else your confidentail data could be available to the next user.

If you are using a pop mail account and if you leave a copy on the server, you would still be able to access those incoming emails via webmail account using a browser, but you would not be able to access 'Sent' emails as they are only on your computer in the Thunderbird profile.

You could put your Thunderbird profile with pop account on an external harddrive and move it between Thunderbird on each computer, but that's not ideal. You would need to use Profile Manager to create a new profile on External drive and then move profile name contents to that drive. Then make sure the default profile on new computer is setup to access and read the profile location. So much easier to use imap mail account or access webmail via a browser.