The concept of the Third Rome is an invention of someone -- off-hand I forget the name -- in 16 century Russia. It assumes that Rome has already fallen, -- physically to the Germanic tribes and theologically to what the East perceives as error of Catholicism, and further to the heresy of Protestantism. The Holy See of the True Church then becomes Constantinople, but that, too, fell to the Turks; it is evident from this (quite prejudiced) observation that Moscow is indeed the Third Rome: the only standing Orthodox empire.
Naturally, Russia historically looked at Constantinople as its historical mission. I don't think any of that is relevant after a hundred years of disintegration in Russia.
The ROC is a rival. They praised the Aglican on their gay ministering moves and yet told them they would not encourage this in their own church.
Very peculiar politics given that any Church out there should be very concerned with gay vectored disintegration.