Consider a finite family D of smooth vector fields on a finite-dimensional smooth manifold M. We define a two-component Markov process (X,A) with state space M x D as follows: Given a starting point on M and an initial vector field from D, the first component X follows the solution trajectory to the corresponding initial-value problem for an exponentially distributed random time. Then, a new vector field is randomly selected from D and X follows the corresponding trajectory for another random time. The second component A records the current driving vector field. We formulate sufficient conditions for uniqueness and absolute continuity of the invariant measure of the Markov semigroup. These consist of a Hoermander-type hypoellipticity condition and a recurrence condition. In the case where M is the real line, we show that invariant densities are smooth away from critical points of the vector fields and present some asymptotics at critical points. The talk is based on work with Yuri Bakhtin and Jonathan Mattingly.