We show that the blind LTI channel estimation problem, when the input sequence is independent, but has time-varying statistics, mimics that for the i.i.d. case under appropriate persistence of excitation conditions. Hence, consistent parametric and non-parametric estimators baaed on a single realization are readily obtained. We establish an ergodicity theorem for the time-averages of non-stationary continuous time processes; we use this to establish blind identifiability of the LTI channel of a filtered inhomogeneous point process, with multiplicative marks. These results extend to o class of time-varying channels as well. The theoretical results are corroborated by simulations.