Motivated by collapsing of Riemannian manifolds and inhomogeneous
scaling of left invariant Riemannian metrics on a real Lie group $G$
with a sub-group $H$, we introduce a family of interpolation equations
on $G$ with a parameter $\epsilon>0$, interpolating hypo-elliptic
diffusions on $H$ and translates of exponential maps on $G$ and
examine the dynamics as $\epsilon\to 0$. When $H$ is compact, we use
the reductive homogeneous structure of Nomizu to extract a converging
family of stochastic processes (converging on the time scale
$1/\epsilon$), proving the convergence of the stochastic dynamics on
the orbit spaces $G/H$ and their parallel translations, providing also
an estimate on the rate of the convergence in the Wasserstein
distance. Their limits are not necessarily Brownian motions and are
classified algebraically by a Peter-Weyl's theorem for real Lie groups
and geometrically using a weak notion of the naturally reductive
property; the classifications allow to conclude the Markov property of
the limit process. This can be considered as "taking the adiabatic
limit" of the differential operators
$\mathcal{L}^\epsilon=(1/\epsilon) \sum_k (A_k)^2+(1/\epsilon)
A_0+Y_0$ where $Y_0, A_k$ are left invariant vector fields and
$\{A_k\}$ generate the Lie-algebra of $H$.