So how do Authorities go about it?
Like "Hey Yomix, please do not mix coins coming from this address, if you refuse to comply, we know where you are and we shall arrest you"

--snip--
Lol, not like that obviously

No mixer will ever respond to authorities because even they may get traced down and these authorities don't like mixers at all if I'm not wrong, and especially they won't respond if the authorities send a message in the way you described.

Just in case, they try to get the specific mixer's email address for support or let's say, PM them here on Bitcointalk, will they ever respond to the authority's concerns?