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We prove here that given a proper isometric action $K\times M\to M$ on acomplete Riemannian manifold $M$ then every continuous isometric flow on theorbit space $M/K$ is smooth, i.e., it is the projection of an $K$-equivariantsmooth flow on the manifold $M$. As a direct corollary we infer the smoothnessof isometric actions on orbit spaces. Another relevant application of ourresult concerns Molino's conjecture, which states that the partition of aRiemannian manifold into the closures of the leaves of a singular Riemannianfoliation is still a singular Riemannian foliation. We prove Molino'sconjecture for the main class of foliations considered in his book, namelyorbit-like foliations.

Smoothness of isometric flows on orbit spaces and applications to the theory of foliations