  The purpose of this paper is to construct a Brownian motion $X :=
(X_t)_{t\geq 0}$ taking values in a Riemannian manifold $M$, together with a
compact valued process $D:= (D_t)_{t\geq 0}$ such that, at least for small
enough ${\mathscr F}^D$-stopping time $\tau> 0$ and conditioned by ${\mathscr
F}_\tau^D$, the law of $X_\tau$ is the normalized Lebesgue measure on $D_\tau$.
This intertwining result is a generalization of Pitman theorem. We first
construct regular intertwined processes related to Stokes' theorem. Then using
several limiting procedures we construct synchronous intertwined, free
intertwined, mirror intertwined processes. The local times of the Brownian
motion on the (morphological) skeleton or the boundary of $D$ plays an
important role. Several examples with moving intervals, discs, annulus,
symmetric convex sets are investigated.
  KEYWORDS: Brownian motions on Riemannian manifolds, intertwining relations,
set-valued dual processes, couplings of primal and dual processes, stochastic
mean curvature evolutions, boundary and skeleton local times, generalized
Pitman theorem.
