There are many undercurrents of the following.
Tribes/Clans in Chechnya are pitted against one another until one dominates and slowly the potential contention switches from internal tribal/ethnic competition to external demands for more resources. The internal dynamics do not go away the populations co-exist in a sense of heirarchy. In Chechnya clans simply had the fall out and Kadyrov clan is on top so there is really not much internal competition as of now. (That won't mean it can't happen in the future.)
Dagestan is a far more interesting case. Lots of ethnicities and even the dominant ethnicities have several clans that could re-align with other ethnicities to try to get at power. What is even more interesting is that prior to radicalization in the region Dagestan was a parliamentary driven state within a state with each ethnicity more or less having pro-rata representation in their parliament. After reforms it became a presidential driven state with more powers transfered over to re-distribute power, which ended up in a skew of it towards dominant ethnicities and clans. This helped radicalize some of the others in addition to other prolifiration from external sources.
In May 2012, the situation nearly spiraled into massive clashes in Makhachkala’s suburbs. Each having interests in Makhachkala’s administration, ethnic Kumyks, Laks, Avars and Chechens clashed. And it took the vigorous effort of the informal leaders of these groups to prevent violence. “Bloodshed was miraculously avoided,” one author wrote following the standoff (Дагестанский информационный еженедельник "Новое Дело" :: Статья "«Угроза целостности Дагестана»" | Выпуск № 20 (1059)).
Between 1994 and 2006, Dagestan’s diversity was
managed through an inter-ethnic cohabitation system,
including representation of the fourteen most important
ethnic groups in an executive body, the Gossovet
(State Council). The system was designed to neutralise
centrifugal forces, as much as confrontations, by
satisfying the aspirations of nationalities. Magomedov
was Gossovet’s president and head of the republic.
Although the introduction of the office of president of
Dagestan was rejected in referendums in 1992, 1993
and 1999, it was finally set up in 2003. However, the
next year Russian President Vladimir Putin abolished
elections for heads of local government throughout
Military perspective of the tinder box.