This paper discusses the time-dependent dynamics of electrostatic vibration energy harvesters (eVEHs) with linear and nonlinear mechanical resonators. These eVEHs are fundamentally nonlinear regardless of whether a linear or nonlinear resonator is being used. The model of the system under investigation has the form of a piecewise-smooth dynamical system of a Filippov type that has a specific discontinuity in the form of a hold-on term. We use a perturbation technique called the multiple scales method to develop a theory to analyze the steady-state dynamics of the system, be it with a linear or a nonlinear resonator. We then analyze the stability of the steady-state orbit to determine when the first doubling bifurcation occurs in the system. This gives an upper bound on the region of steady-state oscillations which allows us to determine a theoretical limit on the power convertible by the eVEH. We then turn our discussion to the nonlinear behavior we see in the system's transition to chaos. Since the eVEH studied here is a Filippov type system, sliding modes and sliding bifurcations are possible in the system. We discuss the evolution of the sliding region and give particular examples of sliding phenomena and sliding bifurcations. An understanding of sliding phenomena is required for analyzing the transition to chaos since segments of sliding motion appear on trajectories that undergo period-doubling bifurcations. The transition to chaos is explained in detail by the example of the system with a linear resonator, however we discuss examples of the system with mechanical nonlinearities and discuss the difference between the linear and nonlinear cases.