We introduce a minimal model of interacting particles relying on conservation of the number of particles and interactions respecting conservation of the center of mass. The dynamics in our model is directly amenable to simple pairwise interactions between particles leading to particle displacements, ensues from this what we call interaction mediated dynamics. Inspired by binary reaction kinetics-like rules, we model systems of interacting agents activated upon pairwise contact. Using Poisson representations, our model is amenable to an exact nonlinear stochastic differential equation. We derive analytically its hydrodynamic limit, which turns out to be a nonlinear diffusion equation of porous medium type valid even far from steady state. We obtain exact self-similar solutions with subdiffusive scaling and compact support. The nonequilibrium steady state of our model in the dense phase displays hyperuniformity which we are able to predict from our analytical approach. We reinterpret hyperuniformity as stemming from correlations in particles displacements induced by the conservation of center of mass. Although quite simplistic, this model could in principle be realized experimentally at different scales by active particles systems.