Thank you, Stephan, for correcting me.
The generalization of Donsker's theorem from $N$-dimensional Euclidean space to general Riemannian manifolds has been worked out by Erik Jørgensen, The Central Limit Problem for Geodesic Random Walks.
The purpose of the present work is to consider the problem of defining the concept of a random walk in a general Riemannian manifold ${\cal M}$, and to investigate the behavior in the limit of a sequence of such random walks. It will be shown that such a sequence, under reasonable assumptions, converges to a diffusion process in ${\cal M}$, and in particular Brownian motion processes will be obtained as limits of sequences of random walks with identically distributed steps. The results which we arrive at in this paper are general versions of well-known classical results concerning the transition from random walks to diffusion processes, for instance: the central limit theorem and Donsker's theorem.