Apologies for upping an old thread, but I'm currently in the middle of something similar.
With regards to the steps quoted above, I've set up the new email at the new host and ready to initiate the domain transfer. However I just want to be absolutely certain that no existing emails will be lost from the old host once it happens. They're using a pop3 account currently and have their email program settings to never delete email from the server (they just do it manually when they fill their max allocation). Is it worth advising them to backup/archive their current emails or will they remain in the defunct inbox once domain transfer is complete?