We provide a detailed and rigorous proof of (a generalized version of) the Ehlers–Geroch theorem on geodesic motion in metric theories of gravity: we assume that (M, g) is a spacetime satisfying an averaged form of the dominant energy condition and some further technical assumptions indicated in the bulk of this paper. Then, a small body which is allowed to deform the original spacetime metric g moves, nonetheless, along a geodesic of (M, g).