The evolution of systems in random media is a broad and fruitful field for the applicaΒ tions of different mathematical methods and theories. This evolution can be characterΒ ized by a semigroup property. In the abstract form, this property is given by a semigroup of operators in a normed vector (Banach) space. In the practically boundless variety of mathematical models of the evolutionary systems, we have chosen the semi-Markov ranΒ dom evolutions as an object of our consideration. The definition of the evolutions of this type is based on rather simple initial assumptions. The random medium is described by the Markov renewal processes or by the semiΒ Markov processes. The local characteristics of the system depend on the state of the ranΒ dom medium. At the same time, the evolution of the system does not affect the medium. Hence, the semi-Markov random evolutions are described by two processes, namely, by the switching Markov renewal process, which describes the changes of the state of the external random medium, and by the switched process, i.e., by the semigroup of operΒ ators describing the evolution of the system in the semi-Markov random medium.