  We investigate effective equations governing the volume expansion of
spatially averaged portions of inhomogeneous cosmologies in spacetimes filled
with an arbitrary fluid. This work is a follow-up to previous studies focused
on irrotational dust models (Paper I) and irrotational perfect fluids (Paper
II) in flow-orthogonal foliations of spacetime. It complements them by
considering arbitrary foliations, arbitrary lapse and shift, and by allowing
for a tilted fluid flow with vorticity. As for the first studies, the
propagation of the spatial averaging domain is chosen to follow the congruence
of the fluid, which avoids unphysical dependencies in the averaged system that
is obtained. We present two different averaging schemes and corresponding
systems of averaged evolution equations providing generalizations of Papers I
and II. The first one retains the averaging operator used in several other
generalizations found in the literature. We extensively discuss relations to
these formalisms and pinpoint limitations, in particular regarding rest mass
conservation on the averaging domain. The alternative averaging scheme that we
subsequently introduce follows the spirit of Papers I and II and focuses on the
fluid flow and the associated 1+3 threading congruence, used jointly with the
3+1 foliation that builds the surfaces of averaging. This results in compact
averaged equations with a minimal number of cosmological backreaction terms. We
highlight that this system becomes especially transparent when applied to a
natural class of foliations which have constant fluid proper time slices.
