The system of kinetic equations for a medium of finely dispersed particles in a rarefied gas is solved by the Chapman-Enskog method. The hydrodynamic transport equations and expressions for the transport coefficients are obtained. The contribution of the three-body interactions to the transport coefficients, determining the first-order corrections in the density expansion, has been calculated for a model of hard spheres. The many-body collision contribution to the dissipative characteristics of the medium is calculated under the condition that the gas-gas collisions in the ternary and higher order collision integrals can be neglected. We also suggest a system of effective kinetic equations which take into account these effects.