We derive kinetic equations describing the low-density (and the large-number-of-particles) limit of interacting particle systems with k-ary interaction of pure jump type supplemented by an underlying "free motion" given by an arbitrary Feller process. The well-posedness of the Cauchy problem, as well as the propagation of the chaos property, is proved for these kinetic equations under some reasonable assumptions. The (spatially nontrivial) Boltzmann and Smoluchowski equations with a mollifier are special cases of our general equations. Our analysis produces new results even for the classical binary models.