On direct and inverse Kolmogorov equations for purely jump-like Markov processes and their generalizations

In the work “On analytical methods in probability theory'” (1931), A.N. Kolmogorov, starting from the relations called Kolmogorov-Chapman equations, derived for transition probabilities of inhomogeneous stochastically defined systems, or, as is now commonly said, for inhomogeneous Markov random processes (in an expanded meaning), reverse and direct equations in the following three cases:
(A) systems with a finite number of states;
(B) systems with countable number of states;
(C) diffusion-type systems with a continuous set of states.
The report, which is largely of a review nature, considers the cases (A), (B) and the purely jump case for a Markov process with a Borel state space.
The report is based on joint work with E.A. Fainberg.