That works only if moved to an online folder.

Thanks. Is this possible to do with iCloud? Apple Mail is automatically synced with iCloud. I guess that it would have to done manually with Thunderbird folders. Is there a link anywhere that describes how to do this?

You have two components. An IMAP account synchronized to the mail server and a Local Folder account on your local drive. Attempts to sync that local folder will almost certainly lead to data loss and corruption. So either leave the mail in IMAP folders within the account or put up with local folders actually being local. There are no other supported synchronization methods.

Thanks. That's what I was afraid of. So to keep my Inbox clear I'll copy any email I want to keep to the local folder on one computer and then move it from the Inbox to the local folder on the other computer. It's a bit of a faff, but not to too bad as long as I don't want to keep too many emails.

you can drag them from Thunderbird to a disk drive and EML files will be saved. Those synchronize across things like Icloud, but re file system files, not Thunderbird mail as such. You can also export the folder periodically and import it as an mbox file using the import export tools. But that is not synchronization.

Mostly I would suggest creating an IMAP account with a provider that offers a lot of free space and just use that for storage instead of local folders. A gmail 15Gb mail account takes a fair bit of filling. If you never use it for anything but an archive type store or advertise it as your mail account it will do the job admirably I would think.