  We study the evolution under their self-gravity of infinite ``shuffled
lattice'' particle distributions, focussing specifically on the comparison of
this evolution with that of ``daughter'' particle distributions, defined by a
simple coarse-graining procedure. We consider both the case that such
coarse-grainings are performed (i) on the initial conditions, and (ii) at a
finite time with a specific additional prescription. In numerical simulations
we observe that, to a first approximation, these coarse-grainings represent
well the evolution of the two-point correlation properties over a significant
range of scales. We note, in particular, that the form of the two-point
correlation function in the original system, when it is evolving in the
asymptotic ``self-similar'' regime, may be reproduced well in a daughter
coarse-grained system in which the dynamics are still dominated by two-body
(nearest neighbor) interactions. Using analytical results on the early time
evolution of these systems, however, we show that small observed differences
between the evolved system and its coarse-grainings at the initial time will in
fact diverge as the ratio of the coarse-graining scale to the original
inter-particle distance increases. The second coarse-graining studied,
performed at a finite time in a specified manner, circumvents this problem. It
also makes more physically transparent why gravitational dynamics from these
initial conditions tends toward a ``self-similar'' evolution. We finally
discuss the precise definition of a limit in which a continuum (specifically
Vlasov-like) description of the observed linear and non-linear evolution should
be applicable.
