Domain transfer

first change the admin-c itself. Do the right thing: record yourself in the admin-C and tech-C, and optionally of the Registrar in the TS.
As for the procedure of domain transfer to another Registrar, then you will ask your new Registrar. Since you did not specify a zone, the exact answer to your question can not be given.

sorry, no Registrar in the TS, and the one who provide work, Nsou domain. Such a thing as a Registrar in General not all DNS-zones exist.

Do specify the zone, maybe everyone has its own rule. but the point is (to logger):
>What is the procedure to transfer a domain to another Registrar? I must ask the current Registrar to have changed >admin-c handle on my nick and after that registrars understand each other?
\r
registrars to understand each other WILL not. you have tech Registrar's request so-called auth code (transfer code) then the new transfer domain, where enter this code, but then they have as you say "agree among themselves", although it all happens automatically
\r
PS: and still call the area, maybe in the TLD — one rules (much more transparent, simpler and more correct), and in local areas their own, where everything is much more complicated

unfortunately .ua is not strong, but it is necessary to read rules nic.ua and many questions will disappear

Many registrars very often enter yourself in admin C, and in Huize writing was in the interests of the client!