We need to transfer domain from another registrar to network solution, when domain transfer is initiated and completed will domain transfer to done with all the details like www,MX, A SPF record or we have to update all these records after domain transfer is complete.
My experience is that if the nameservers point away from the registrar i.e. the DNS is hosted elsewhere, then a change of registrar does not affect DNS at all.
All the registrar holds on you is your account information and name server entries, your DNS records are hosted by your DNS host which in many instances is the same as your web host. Just transfer the domain name and give them your name server addresses and within 48 hours it should be fine.