Moderates isn't necessarily the word I would use. By Clan standards, they are quite out there, though they limit themselves to graffitti and the like rather than bombing campaigns.
The Clans probably would persecute them if they could, but like the Scientist conspiracy before them, they are a closed group. Very difficult to find, and they operate in cells.
The movement's goal is to do away with the Clans and the Caste system in favour of a society that basically inverts the Clan social structure- warriors at the bottom, civillians on top. It is popular with the Labourer and Technician Castes, primarily because they lack the power or prestige of the other two.
The goal of the leader, the uKhan (under-Khan) is to gather as much temporal power to himself as possible. The ideology does not interest him; he merely wants to be in charge and chafes at not being there yesterday.
The movement only really surfaces on Vega, and even then because it's the Bears' Omega Galaxy- which appears to having gone from a unit the Clan at large finds distasteful to an active sink unit, at least outside the ranks of the Warriors. Because they have greater autonomy from the Bears by virtue of not being in the Dominion and mingling more freely with Spheroids, the lower castes find the new ideas interesting.
It is worth nothing that perhaps not all of their documentation is forged. The uKhan gets his manifesto forged, claiming that Kerensky had always planned for the Caste system to break down after the passage of a set period of time, after which it is rainbows and puppy dogs for all. However, he did have some authentic writings of Kerensky, and it is implied in the novel that mainstream Clan society would not be too fond of what it said about the Castes or what Kerensky's intention for them and the Clans were.
Apparently whatever it did say did not go far enough for the uKhan's tastes, though.
As to how pervasive it is? We only really know about it in the Bears. The Wolves have always been generous with their civilians, as have the Diamond Sharks/Sea Foxes. The Hell's Horses and Jade Falcons are somewhat more repressive, and if it is to be found outside the Bears those are the Clans I would look to.