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We study the effect of coarse graining the dynamics of a pressureless self-gravitating fluid (coarsegrained dust) in the context of cosmological perturbation theory, in both the Eulerian and Lagrangian frameworks. We obtain recursion relations for the Eulerian perturbation kernels of the coarse-grained dust model by relating them to those of the standard pressureless fluid model. The effect of the coarse graining is illustrated by means of power and cross spectra for the density and velocity, which are computed up to oneloop order. In particular, the large-scale vorticity power spectrum that arises naUirally from a mass-weighted velocity is derived from first principles. We find qualitatively good agreement for the magnitude, shape, and spectral index of the vorticity power spectrum with recent measurements from N-body simulations and results from the effective field theory of large-scale structure. To lay the ground for applications in the context of Lagrangian perturbation theory, we finally describe how the kernels obtained in Eulerian space can be mapped to Lagrangian ones. [ABSTRACT FROM AUTHOR]