Filtered shot noise processes have proven to be very effective in modeling the evolution of systems exposed to shot noise sources and have been applied to a wide variety of fields ranging from electronics through biology. In particular, they can model the membrane potential V(m) of neurons driven by stochastic input, where these filtered processes are able to capture the nonstationary characteristics of V(m) fluctuations in response to presynaptic input with variable rate. In this paper we apply the general framework of Poisson point processes transformations to analyze these systems in the general case of nonstationary input rates. We obtain exact analytic expressions, as well as different approximations, for the joint cumulants of filtered shot noise processes with multiplicative noise. These general results are then applied to a model of neuronal membranes subject to conductance shot noise with a continuously variable rate of presynaptic spikes. We propose very effective approximations for the time evolution of the V(m) distribution and a simple method to estimate the presynaptic rate from a small number of V(m) traces. This work opens the perspective of obtaining analytic access to important statistical properties of conductance-based neuronal models such as the first passage time.