  The paper deals with a certain class of random evolutions. We develop a
construction that yields an invariant measure for a continuous-time Markov
process with random transitions. The approach is based on a particular way of
constructing the combined process, where the generator is defined as a sum of
two terms: one responsible for the evolution of the environment and the second
representing generators of processes with a given state of environment. (The
two operators are not assumed to commute.) The presentation includes fragments
of a general theory and pays a particular attention to several types of
examples: (1) a queueing system with a random change of parameters (including a
Jackson network and, as a special case: a single-server queue with a diffusive
behavior of arrival and service rates), (2) a simple-exclusion model in
presence of a special `heavy` particle, (3) a diffusion with drift-switching,
and (4) a diffusion with a randomly diffusion-type varying diffusion
coefficient (including a modification of the Heston random volatility model).
