Not quite sure what way you’re going about this…
You don’t need to transfer any domains. They can stay wherever they are at the moment, regardless of your web hosting.
What you need to do is point those domains to your new server/web host.
This can take up to 48 hours to propagate around the world – up to a week with a small number of more unusual ISPs – but generally within a few hours.
If the website is ready set up on both servers (i.e. old and new) the changes will be seamless and you won’t need to touch the MX records for your email.
But if you prefer to transfer the domains, you can…you will still need to do all of the above.
Edit: Mind you, 1and1, if I remember rightly, are utter bar stewards and like to charge you for every change you make. So it might not be a bad idea transferring. No experience of TSO myself, but I can recommend 123-reg for domains only.