The Grote-Hynes transmission coefficient for the rate of barrier crossing in the presence of memory friction is rederived here using the method of reactive flux. By combining the methodology developed in an earlier paper [D. J. Tanner and D. Kohen, J. Chem. Phys. 100, 4932 (1994)] with the non-Markovian Fokker-Planck equation of Adelman [S. Adelman, J. Chem. Phys. 64, 124 (1976)] we are able to obtain not only the asymptotic rate constant but the behavior of the rate constant at all times. The salient features of the time dependent rate constant, k(t), are interpreted in terms of the time evolution of the representative distribution functions that originate at the top of the barrier. The short time behavior of the rate constant is very different in the dynamic and static limits, with close analogies to the stochastic theory of spectral line shapes. The dependence of the "plateau time"-the time for the rate constant to reach its steady state value-on the memory kernel is explored numerically, and analytical expressions are obtained in the dynamic and static limits.