You must first purchase a new hosting account with a provider(DialWebHosting) whose mail server supports IMAP
that you use to receive your emails. You then should point your domain to the new hosting account by making the required changes in the name servers. Check with the support team what nameservers you must give to the domain to enable it to point to the new server. Once done, the DNS propagates the name change across the internet. It may take anything between 48 hours to 96 hours for the change to take effect.
Next, you must set up IMAP accounts in Outlook one on the account with your server and one for your new server. You must be ready with all the details such as email account password, IP addresses of your incoming mail (IMAP/POP3) server and outgoing SMTP
server, port numbers etc.
Once the two IMAP set up is completed, simply drag and drop the emails from one inbox folder to another. Do not try to move all the emails at one time. Drag/drop them one at a time. The time taken will depend on your internet connection speed.
Maintain both accounts for the propagation period up to 96 hours. Then you may delete your old account.