This paper deals with Cauchy matrix family of a linear differential equation with right part dependent on a step Markov process and an impulse type dynamical system switched by the above process. All the above mentioned stochastic dynamical objects are also dependent on small positive parameter ε, and the infinitesimal operator of Markov process is proportional to ε−1. This means that impulse dynamical system is rapidly switched and one may simplify this using merger procedur to Markov process and averaging procedures to impulse dynamical system and matrix evolution family. Applying these procedures one achieves more simple linear differential equation for matrix evolution family, which becomes now dependent on more simple dynamical systems such as an ordinary differential equation with a right part switched by a lumped Markov process. It is proved that under some hypotheses one may successfully apply these resulting evolution families not only to approximation of the initial family on an arbitrary finite time interval but also to describe a time asymptotic of it.