The Jihadi Insurgency and the Russian Counterinsurgency in the North Caucasus, by Gordon M. Hahn. Post-Soviet Affairs, Volume 24, Number 1 / January-March 2008.
Abstract
A political scientist reviews the vicissitudes of the Caucasus jihadi insurgency and Russia's counterinsurgency efforts since 2005, drawing on the press, jihadist websites, and scholarship on insurgencies and counterinsurgencies. The development of jihadi-oriented fighting units, the rise and decline of Maskhadov's supremacy, the rise of Islamist elements, the Chechen separatist movement and the expansion of the jihad, Moscow's counter-jihadi successes, the jihadi network after the death of Basayev, and the rise and career of Ramzan Kadyrov are examined. These events are analyzed in terms of the dynamics of insurgency and counterinsurgency conflict.
39 Page pdf at the link.

I've been looking for examinations of Russian counterinsurgency in the Kadyrov era. Came across this and thought I would share it.