  Many physical observables can be represented as a particle spending some
random time within a given domain. For a broad class of transport-dominated
processes, we detail how it is possible to express the moments of the number of
particle collisions in an arbitrary volume in terms of repeated convolutions of
the ensemble equilibrium distribution. This approach is shown to generalize the
celebrated Kac formula for the moments of residence times, which is recovered
in the diffusion limit. Some practical applications are illustrated for
bounded, unbounded and absorbing domains.
