Generally it takes hours (minimum 8-24)) for MX records or any dns records to update and propagate worldwide. If your user send an email during propagation period it will certainly reach at old host. The best option to recover the email from old host to download the emails by configuring it into any email client or if you are able to access it using webmail then forward it to your other email address.
Thanks, it looks like the migration itself went off without a hitch. The old host's emails are accessible through web mail, and the new host's emails are accessible normally.
I gather that "IMAP transfer" is a technique that we can use to move emails from the old host to the new one. That would be very useful, but I haven't heard of it before. I tried googling the term but I didn't find any references that seemed to fit. Can you tell me more about it, please?
Usually one sets the MX records on the old host to point to the new host. The trick with doing this during a move is, you can't use the domain name for the new host, you have to use the actual hostname of the server, which you can find by looking up the IP for your domain name, then looking up that IP to see what it returns.
For instance, if your domain name is yourdomainname.com you would look up yourdomainname.com and find it has an ip of 172.16.32.56.
You would then look up that IP (172.16.32.56) and see that has a hostname of server53.com.
Then you'd set your MX record on the old host to point to server53.com.
With "IMAP transfer", you can connect to both hosts simultaneously (you may have to use the server name for the old host) and you can then drag and drop from one server to the other. Not at all an official term or definition, hence why you didn't find it via Google, but nevertheless a handy trick.