I understand that the OP's original focus is classical statistical mechanics. However, i think that the question is of interest from a more general viewpoint including the dynamical systems/integrability and/or the quantum statistical mechanics point of view.  
In this sense, i am not sure if ths is the kind of answer you are hoping for, but if you are 
>interested in cases where the higher-order interactions are instead a key ingredient of the story

then you should look into the field of diffractive scatering models. That is, many-body models, non-integrable in the Bethe ansatz sense, not necessarily preserving the total number of particles and with the two-particle scattering matrices violating the Yang-Baxter equation:  
In **non-diffractive scattering**, many-body collisions factorise in a sequence of two-body collisions; these are generally elastic collisions, the matrices satisfy the Yang-Baxter eq and we have complete integrability in the Bethe ansatz sense. These generalize -in a sense- the Boltzmann eq models you are mentioning. They preserve the number of particles and furthermore the set of outgoing momenta is the same with the the set of the incoming momenta. (some particles exchange velocities during these collisions). Binary collisions dominate the behaviour.  
In **diffractive scattering**, all the above are violated. Only energy and momentum conservation survive. Higher order (that is many particle) interactions dominate and in some cases they seem to be necessary in order to comply with experimental evidence of cooling down. See for example [Diffractive scattering of three particles in one dimension: a simple result for weak violations of the Yang--Baxter equation][1], where a proposal for the formulation of a kinetic theory with three-particle collisions is suggested. (see especially the conclusions section of the paper).  
(I think this should be studied in conjuction with [A rigorous derivation of a ternary Boltzmann equation for a classical system of particles][2] proposed in the comments by user Matt Rosenzweig). 


  [1]: https://arxiv.org/abs/1211.4110
  [2]: https://arxiv.org/abs/1903.04279