Each service uses IMAP slightly differently. By default, many may download your inbox but no other folders. Therefore, make sure that Thunderbird gets all your messages every time it looks for emails on the server.
However, the best method is hidden in Thunderbird's configuration editor. Go to Thunderbird's Options and go to Advanced> Configuration Editor and click on the "I Accept the Risk" button when prompted. Find the setting mail.server.default.check_all_folders_for_new and double-click it to set it to True.
From now on, Thunderbird should automatically sync your entire account, not just your Inbox. If this is not the case, after adding a new account, you may need to quit and restart Thunderbird or wait for messages to be retrieved for each folder. This process can take a long time if you have a lot of old messages. So do other jobs and let him do his thing Make sure your messages are readable. Again, different web services can implement IMAP slightly differently. So if you can not view your messages offline, you may need to do a few days of searching and testing to make sure everything is working properly.
Once you're done, you can be sure that Thunderbird will download all your emails. Just keep it running in the background while using your computer. If you do not want to waste resources, set the Windows Task Scheduler to start Thunderbird once or twice a day to update the local database with the latest messages. If you ever lose the Internet, just start Thunderbird and it will run in offline mode so you can access your downloaded archive to stay productive.