[en] We study the noisy nonequilibrium dynamics of a conserved density that is driven by a fluctuating surface governed by the conserved Kardar–Parisi–Zhang equation. We uncover the universal scaling properties of the conserved density. We consider two separate minimal models where the surface fluctuations couple (i) with the spatial variation of the conserved density, and (ii) directly with the magnitude of the conserved density. Both these two models conserve the density, but differ from a symmetry stand point. We use our result to highlight the dependence of nonequilibrium universality classes on the interplay between symmetries and conservation laws. (paper: classical statistical mechanics, equilibrium and non-equilibrium)