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We investigate effective equations governing the volume expansion ofspatially averaged portions of inhomogeneous cosmologies in spacetimes filledwith an arbitrary fluid. This work is a follow-up to previous studies focusedon irrotational dust models (Paper I) and irrotational perfect fluids (PaperII) in flow-orthogonal foliations of spacetime. It complements them byconsidering arbitrary foliations, arbitrary lapse and shift, and by allowingfor a tilted fluid flow with vorticity. As for the first studies, thepropagation of the spatial averaging domain is chosen to follow the congruenceof the fluid, which avoids unphysical dependencies in the averaged system thatis obtained. We present two different averaging schemes and correspondingsystems of averaged evolution equations providing generalizations of Papers Iand II. The first one retains the averaging operator used in several othergeneralizations found in the literature. We extensively discuss relations tothese formalisms and pinpoint limitations, in particular regarding rest massconservation on the averaging domain. The alternative averaging scheme that wesubsequently introduce follows the spirit of Papers I and II and focuses on thefluid flow and the associated 1+3 threading congruence, used jointly with the3+1 foliation that builds the surfaces of averaging. This results in compactaveraged equations with a minimal number of cosmological backreaction terms. Wehighlight that this system becomes especially transparent when applied to anatural class of foliations which have constant fluid proper time slices.

On average properties of inhomogeneous fluids in general relativity III: general fluid cosmologies