How to keep old IMAP emails when changing host

Mail/Mailbox/New Mailbox - create an On My Mac mailbox and copy the e-mails there for storage while the transition is made.

If you are changing hosts, the old account will probably have the wrong server settings and won't work.

I would copy the e-mails to the new mailbox, delete the old account, and then set the account up again on the new host.

Copying your mail to local disk is easy. First you create a local mailbox to hold your archival mail. Then you can copy or move mail from the server to that location. Here are the steps:

Create a local mailbox
Start Apple Mail
From the menu select Mailbox > New Mailbox...
Result: The "New Mailbox" window appears
For Location select "On My Mac"
For Name enter a name for your new local mailbox
Click OK

Result: Your new mailbox appears in the mailbox listing
Move mail to the local mailbox
Open the mailbox on the server that contains the message or messages you'd like to copy or move
Select the messages you wish to copy; to select all the messages in a mailbox use command-a (command icon - a)
Result: The messages to be copied are highlighted
ctrl-click on the highlighted messages to see a menu of options and select "Copy To" or "Move To" (depending on whether you want to copy or move the messages)
Then choose the folder you created in previously from the pop-up menu

Result: The messages you selected are copied to the folder on your local machine
If you selected "Copy To" then the original messages are still on the mail server and count against your Email Quota. For more information on staying within your email quota, see How do I manage my email quota? If you selected "Move To" the messages will disappear from the mail server and will no longer count against your email quota.

Repeat the steps as needed to copy or move mail from multiple mailboxes on the server to your preferred set of mailboxes on your local machine.