Let us begin with a Lagrangian of the form 

$$\mathscr L= \frac 12 \sqrt{-g}g^{\mu\nu}\partial_\mu\phi(x)\partial_\nu\phi(x)+\mathscr L_g,$$

where $$\mathscr L_g=\frac 1{16\pi k}\sqrt{-g}R.$$  Suppose as well that there are no Killing vectors associated to the metric $g^{\mu\nu}$ except for, say, a global timelike Killing vector if it helps the argument.

Associated to $\mathscr L$ is a locally conserved set of 10 currents from the Poincare group:

$$T^{\mu\nu}=\partial^\mu\phi(x)\partial^\nu\phi(x)-g^{\mu\nu}\mathscr L$$

for each spacetime translation, and $\epsilon_{\alpha\beta}x^\alpha T^{\mu\beta}$ for each spacetime rotation. 

Locally we have $$\nabla_\mu T^{\mu\nu}=0$$ so these quantities are conserved only locally.  

My question is, what is the obstacle to patching these locally conserved quantities together to make a globally conserved quantity:

$$Q=\int T^{0 \nu}f_\nu \;d^3x$$

with 

$$dQ/dt=0$$

where $f_\nu$ might be a gluing function connecting the momentum flowing out of one patch of infinitesimal volume and into another?