we explain limit theorems associated with a family of random ordinary differential equations on manifolds, driven by randomly perturbed vector fields. after rescaling, the differentiable random curves converge to a markov process whose markov generator can be written explicitly in hormander form. we also give rates of convergence in the wasserstein distance. example 1. a unit speed geodesic, which chooses a direction randomly and uniformly at every instant of order $1\epsilon$, converges to a brownian motion as epsilon tends to 0. furthermore their horizontal lifts converge to the horizontal brownian motion. examples 2. inspired by the problem of the convergence of berger's spheres to a $s ^ {1/2}$, we introduce a family of interpolation equations on a lie group $g$. these are stochastic differential equations on a lie group driven by diffusion vector fields in the direction of a subgroup $h$ rescaled by $1\epsilon$, and a drift vector field in a transversal direction. if there is a reductive structure, we identify a family of slow variables which, after rescaling, converges to a markov process on $g$. furthermore, the projection of the limiting markov process to the orbit manifolds $g/h$ is markov. the limits can be identified in terms of the eigenvalue of a second order differential operator on the subgroup and the $ad(h)$ invariant decomposition of the lie algebra.