The Zwanzig model for the kinetics of escape through a fluctuating bottleneck is generalized to handle fluctuations that can be described by a non-Markovian Gaussian process. Our formulation is based on a multidimensional Fokker-Planck equation with a quadratic sink term. It is shown that the calculation of the time dependence of the survival probability can be reduced to a matrix eigenvalue problem. The elegant Wang-Wolynes expression, first derived using path integral techniques, for the effective rate constant that determines the asymptotic behavior of the survival probability, is recovered. When the fluctuations are described by Langevin dynamics (i.e., diffusion in position and velocity space) on a harmonic potential, analytical expressions for the survival probability and effective rate constant for all values of the friction, are derived. As a function of the friction constant, these exhibit "Kramers turnover" behavior.