  We present an abstract framework for establishing smoothing properties within
a specific class of inhomogeneous discrete-time Markov processes. These
properties, in turn, serve as a basis for demonstrating the existence of
density functions for our processes or more precisely for regularized versions
of them. They can also be exploited to show the total variation convergence
towards the solution of a Stochastic Differential Equation as the time step
between two observations of the discrete time Markov processes tends to zero.
The distinctive feature of our methodology lies in the exploration of smoothing
properties under some local weak H\"ormander type conditions satisfied by the
discrete-time Markov processes. Our H\"ormander properties are demonstrated to
align with the standard local weak H\"ormander properties satisfied by the
coefficients of the Stochastic Differential Equations which are the total
variation limits of our discrete time Markov processes.
