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.