Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $\sum_kY_kα_k(z_t^ε(ω))$ where $Y_k$ are vector fields, $ε$ is a positive number, $z_t^ε$ is a ${1\over ε} {\mathcal L}_0$ diffusion process taking values in possibly a different manifold, $α_k$ are annihilators of $ker ({\mathcal L}_0^*)$. Under Hörmander type conditions on ${\mathcal L}_0$ we prove that, as $ε$ approaches zero, the stochastic processes $y_{t\over ε}^ε$ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.