The emails are in different formats depending on the server, and it can be very complicated to migrate such as migrating a website.

The best and simple solution is this one explained hereafter

Once you've changed the DNS of your domain and point to us :

1. Create same email accounts from your cPanel :
   https://support.yoorshop.hosting/knowledgebase/5471/Create-email-account-in-cPanel.html
2. Configure your email software these new email accounts as pop or imap, following well the informations given in email account in cPanel, see the button 'Settings' or from cPanel V70 : 'Connected devices'

(Please do not change the settings of existing accounts mails in your email software with our settings, this will lose your emails)

3. From your email software, do a drag and drop of emails from the old account to the new account, and wait, this can take time depending on the volume ...

NB :
POP only store locally your emails
IMAP synchronizes emails in both ways : your computer-server