The subject of cosmological backreaction in general relativity is often approached by coordinate-dependent and metric-based analyses. We present in this letter an averaging formalism for the scalar parts of Einstein’s equations that is coordinate-independent and only functionally depends on a metric. This formalism is applicable to general 3  +  1 foliations of spacetime for an arbitrary fluid with tilted flow. We clarify the dependence on spacetime foliation and argue that this dependence is weak in cosmological settings. We also introduce a new set of averaged equations that feature a global cosmological time despite the generality of the setting, and we put the statistical nature of effective cosmologies into perspective.