Transfering registrars doesn’t have anything to do with where your domain is physically pointed, though you may have some intermittent downtime while the new registrar resolves.
Now, changing the nameservers is something else entirely. And yes, you probably will lose your email if you don’t take steps to back it up.
If your email is all POP3, then it’s already on your local computer as soon as you read it, so really you’d just have to worry about your email in the downtime before the domain starts resolving to DH.
If it’s IMAP, that’s more problematic. Here’s what I did. I had a second domain, so I set up email addresses for all my existing email addressses on the second domain. Then using your mail client, connect to both IMAP servers. You can drag and drop your email from the old server to the new server. Then, once your old domain is resolving to DH, repeat the process in reverse. (ie, set up your email addresses and copy your email from the second domain back to the original one that should be resolving to the new server.)
I hope that helps.
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