We propose a new model for the description of a gravitating multiparticle system, viewed as a kinetic gas. The properties of the (colliding or noncolliding) particles are encoded into a so-called one-particle distribution function, which is a density on the space of allowed particle positions and velocities, i.e., on the tangent bundle of the spacetime manifold. We argue that an appropriate theory of gravity, describing the gravitational field generated by a kinetic gas, must also be modeled on the tangent bundle. The most natural mathematical framework for this task is Finsler spacetime geometry. Following this line of argumentation, we construct a coupling between the kinetic gas and a recently proposed Finsler geometric extension of general relativity. Additionally, we explicitly show how the general covariance of the action of the kinetic gas on the tangent bundle leads to a novel formulation of its energy-momentum conservation in terms of its energy-momentum distribution tensor.