The purpose of this paper is to provide equations to model the evolution of effective diffusion over a Riemannian fiber bundle (under the hypothesis of infinite diffusion rate along compact fibers). These equations are obtained by projecting the diffusion equation onto the base manifold of the fiber bundle. The projection (or dimensional reduction) is achieved by integrating the diffusion equation along the fibers of the bundle. This work generalizes an put into a general framework previous work on effective diffusion over channels and the interfaces between curved surfaces.