Migrating hosted domains and email accounts between 2 dedicated servers

I hope this is the most appropriate section to ask this;
I am looking for the best method to manage the following situations :
I am hosting domains on an old, obsolete and slow dedicated machine at provider1, and have to migrate those domains with their email accounts to a brand new powerful server at provider2. Both machines run Plesk Linux and the most recent one has a good migration tool that makes transfering data a rather quick and easy process. All my domains are managed by a third-party, very reliable and quick DNS service.
My question however, is how can I make that change as smooth and quick as possible for email users, knowing that ideally those persons should not be aware of the change, only they'll realize current email issues will be gone.
All email servers are in the form "theirname.com" so when the main domain for webhosting is transfered, so is their incoming mail server.
I do have a way to very quickly update the DNS records from the old machine to the new one, once data is transfered, however I am sure some providers will take at least 24 hours to update their own records, so emails directed to the uses during that time would still go to the old machine ?
What would the best method be so that no one can immediately have their email client read from the new machine, while email would still arrive at the old location's pop accounts, that would therefore not be read from anymore ?
Would changing the domains TTL before that migration help ?

Thanks for your attention on what I believe must be quite a common issue among people providing hosting services and having to upgrade hardware ?
Best regards.