If the new server will not have the messages, you need to download everything you want saved. If POP, you already have the messages on your PC. If IMAP, I suggest 1) checking the synchronization&storage section to confirm the desired folders are authorized for download, and 2) click file>offline>downlload to ensure that all messages, not just headers are downloaded. Next, copy the folders to Local Folders to ensure not losing the messages. Only after doing that would you close the old account. And, be sure to do backups afterwards.

Thank you for the response. That would keep the messages, which is basically what I have done, but I want all the messages for a particular email to be displayed, from the old and the new server. The question is then, how do I stop Thunderbird "syncing", all I want it to do is to download messages that are not on the local machine, I don't want it to delete messages from the machine that are not on the server, nor to delete messages from the server that are not on the machine, I don't want it to delete anything, that is why I ticked the choice not to delete anything. Obviously my understanding of what that does is not Thunderbird's :)

Outlook Express did just what was needed in 1998, the last 22 years I have only used webmail, and started using Thunderbird beginning of this year as I need the emails on the machine.

I think the only way is for me to change to POP, download all from the old server, then change the connection details to the new server also using POP, then I should retain the messages from the old server and the messages from the new server will just add on to those already there.

Means creating lots of new accounts, as you can't change the server type.

Downloading IMAP as I suggested accomplishes the same as setting up a POP account. The download from IMAP does not duplicate anything; it just ensures that all messages are saved. Unless I misunderstood, there is nothing else to do for the old account, since you have the downloaded messages and can save them in Local Folders.

If I have them saved in local folders, can I then view them in the "new" mailbox (same email address but on another server) along with the new messages that will come in to the new server? As I understand it, and what happened to me was that as soon as I refreshed from the new server, all the old emails previously downloaded from the old server were deleted because they do not exist on the new server.

It seem you have an IMAP account. IMAP accounts have messages on the server, providing a copy on PC, but the sync process does as you say, keeping the PC in line with the server. You can keep the old messages in Local Folders and read and reply from there. If you want them as part of the account's inbox, you would need to upload them to the inbox, a problematic venture that does not always work. To do it, I suggest uploading small groups at a time.

Yes, that is exactly it, I thought the IMAP was indicated in my first post. However creating new POP accounts on the old server, downloading all the mail from the old server, then editing the server setup of the account to the new server works just fine. All emails old and new are then in the same place. Setting the leave message on server until I delete and backing up the whole mail folder automatically will keep things safely, and will also mean that if I ever want to change my hosting and email again, I won't have the same problem. This will work for me as I only use a PC for email, I don't look at it on any other devices, and for others they can use webmail on portable devices for new mail after the change over, old mail they would have to check on their PCs, a minor inconvenience, which will grow less as time moves on.