A general theory is presented for the thermally activated rate constant in systems influenced by spatially dependent and. time correlated friction. The theory is valid at all damping strengths and goes uniformly from the energy diffusion limit to the spatial diffusion limit. Results of the theory for a model system with an exponentially time correlated and spatially dependent friction kernel are compared with results from a numerically exact solution of the equivalent generalized Langevin equation. Predictions of the theory are found to be in excellent agreement with the numerical simulation results. The phenomenon of memory suppression of the rate is observed for long time scale frictions and its modification due to the spatial dependence of the friction is discussed. The effects of spatially dependent friction can be understood through a quantity called the "average spatial modification" of the coupling between the reaction coordinate and the environment.