Thermally activated escape over a fluctuating barrier is studied by numerical simulations of the Langevin equation. A class of merged harmonic oscillator potentials are used to model barrier fluctuations continuously over the entire domain of the dynamics. Though the friction constant in the Langevin equation is Ohmic (local), the correlation in the stochastic potentials is taken to be exponential (nonlocal) with a specified decay constant. These models have previously been restricted to the overdamped regime. In this limit, the high friction constants ensure the system's thermalization despite the additional forces acting on the system effected by the stochastic potential of mean force. To insure that equipartition is obeyed below the overdamped regime, the friction constant is modified so as to dissipate this excess energy. This phenomenological choice is exact if the fluctuations in the stochastic potential are induced by a Gaussian random force and are either uniform across the configuration space or the dynamics are determined only at the transition state. Otherwise it provides agreement with the exponent of the equilibrium distribution to second order in the inverse temperature. With this scheme, the stochastic dynamics are shown to exhibit both the Kramers turnover and resonant activation over a wide range of friction and decay constants. The zero and infinite limits of the latter are also obtained using both simulations and analytic rate formulas.