But if you are talking about moving over just dns servers (not the records that are in those dns servers) that is automatically ported over when you change registrars. But if the dns servers are the dns servers of the registrar that isn't what you want. Which is one of the reasons to use third party dns. If you switch registrars the gaining registrar automatically sets the primary and secondary dns to what it was at the old registrar.
I wrote this fast but if you have any questions let me know. This is what I do.
To answer about mail server depends on whether you mean the mail server will run on your equipment or on a VPS or?