  We consider generic Lemaitre-Tolman-Bondi (LTB) dust models to probe the
gravitational entropy proposals of Clifton, Ellis and Tavakol (CET) and of
Hosoya and Buchert (HB). We also consider a variant of the HB proposal based on
a suitable quasi-local scalar weighted average. We show that the conditions for
entropy growth for all proposals are directly related to a negative correlation
of similar fluctuations of the energy density and Hubble scalar. While this
correlation is evaluated locally for the CET proposal, it must be evaluated in
a non--local domain dependent manner for the two HB proposals. By looking at
the fulfillment of these conditions at the relevant asymptotic limits we are
able to provide a well grounded qualitative description of the full time
evolution and radial asymptotic scaling of the three entropies in generic
models. The following rigorous analytic results are obtained for the three
proposals: (i) entropy grows when the density growing mode is dominant, (ii)
all ever-expanding hyperbolic models reach a stable terminal equilibrium
characterized by an inhomogeneous entropy maximum in their late time evolution;
(iii) regions with decaying modes and collapsing elliptic models exhibit
unstable equilibria associated with an entropy minimum (iv) near singularities
the CET entropy diverges while the HB entropies converge; (v) the CET entropy
converges for all models in the radial asymptotic range, whereas the HB
entropies only converge for models asymptotic to an FLRW background. The fact
that different independent proposals yield fairly similar conditions for
entropy production, time evolution and radial scaling in generic LTB models
seems to suggest that their common notion of a "gravitational entropy" may be a
theoretically robust concept applicable to more general spacetimes.
