My understanding (which I think is correct but I'm not guaranteeing it) is that if Outlook is configured to use IMAP for syncing between your email provider's server and it's own local storage, the emails will still be on the remote server and will therefore download into the Inbox of any other email client (such as Thunderbird) that you might use in the future. However they'll only be in folders if those folders also exist on the remote server (and not just locally in Outlook). Assuming that you're using IMAP, you can view which folders (if any) exist on the remote server by following the instructions here
https://support.microsoft.com/en-us/office/choose-which-imap-account-folders-appear-in-outlook-32103602-a568-4499-9d25-54fe236aee6a
or, possibly simpler, by bypassing Outlook altogether and logging into your email account (using your preferred web browser) through the provider's webpage.
However if you've got Outlook configured to use the older POP3 protocol for transferring emails, then (unless you've changed the default settings) mail will automatically be deleted from the remote server when you use Outlook to transfer it to local storage on your computer.
The more I think about it now, the easiest way to answer your question is probably to log into your email account using webmail (as at the end of my first paragraph). Whatever you see there should be accessible, via IMAP, if you decide to use a different email client. i.e. if there's only an inbox, with no folders, that's what will get forwarded to the new client. If the folders are present, they should be forwarded to. If the inbox is empty, you've almost certainly been using POP3, rather than IMAP, meaning that there's nothing available to transfer to a new email client.
However you should also remember that it's usually possible to export all of your emails from Outlook into another client anyway, avoiding the risk of finding out whether they can be downloaded again from the remote server. For example, the instructions for getting email from Outlook imported into Thunderbird can be found here:
https://support.mozilla.org/en-US/kb/switching-thunderbird#w_importing-from-outlook-and-eudora