  We prove existence and uniqueness of solutions to Fokker--Planck equations
associated to Markov operators multiplicatively perturbed by degenerate
time-inhomogeneous coefficients. Precise conditions on the time-inhomogeneous
coefficients are given. In particular, we do not necessarily require the
coefficients to be neither globally bounded nor bounded away from zero. The
approach is based on constructing random time-changes and studying related
martingale problems for Markov processes with values in locally compact,
complete and separable metric spaces.
