Integral form of energy-momentum tensor conservation (Stokes' theorem)

Is there a way in which the conservation law of the energy momentum tensor $nabla _nu T^{munu}=0$ can be written in integral form using Stokes’ theorem, namely as something roughly similar to:

$$ 0 = int_Omega d^4x sqrt{-g}nabla _nu T^{munu} = int_{partial Omega} ? $$

In particular i’m imagining a case where we choose the region $Omega$ such that $T^{munu}$ vanishes on the boundary except for some "initial" and "final" spacelike slices as in this image:

so what we get is a relation between the initial and final spatial integrals.
In flat spacetime I guess this just gives the expected conservation of the total 4-momentum, if we choose the slices to be at constant time coordinate $t$ :

$$ int_{V_1} d^3x T^{0mu} – int_{V_0} d^3x T^{0mu} = 0 $$

but what is the general version of this ? i.e. what is the equivalent relation when (a) spacetime is not flat or (b) the integration region is arbitrary?