As far as I understood, you have created the same email accounts on the new server that your old server has. As soon as it is done, you may update your NS. This will point visitors of your domains and mails sent to your email addresses associated with these domains to the new servers. Thus, the new messages will arrive to the new server, while the messages on the old server (in case you use IMAP or POP3 with "leave messages on the server" option enabled in your email client) will stay there. If you would like to save old messages, you may download them to your email client using POP3 protocol or use appropriate backup option available in your Cpanel.

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