Every service uses IMAP a little differently. By default, many may download your inbox, but not other folders, so you’ll want to make sure Thunderbird is grabbing all your messages every time it checks for mail on the server.
There are a few ways to accomplish this, but the best method is hidden in Thunderbird’s Config Editor. Head to Thunderbird’s Options and go to Advanced > Config Editor, and click the “I Accept the Risk” button when prompted. Search for the mail.server.default.check_all_folders_for_new setting and double-click it to set it to “True.”
From now on, Thunderbird should automatically sync your entire account, not just your Inbox. If it doesn’t, you may need to quit and restart Thunderbird after adding a new account, or wait for it to grab messages for every folder. Again, this process may take a long time if you have a lot of old messages, so continue doing other work and let it do its thing.
When it’s done, put your PC in Airplane Mode and poke around your email folders to see if your messages are readable. Again, different web services may implement IMAP slightly differently, so if you can’t view your messages offline, you may need to do a little digging and testing over the course of a few days to get everything working properly.
Once you’re confident Thunderbird is downloading all your mail, just let it run in the background as you use your computer. If you prefer not to waste resources, set Windows’ Task Scheduler to launch Thunderbird once every day or two so it keeps its local database up to date with the latest messages. If you ever lose internet, just start up Thunderbird and it’ll run in Offline Mode, allowing you to access your downloaded archive so you can stay productive.