Let us consider the case of the stress energy tensor field $T_{\mu\nu}(x)$ in Minkowski spacetime $M^4$. Let $v$ be a constant co-vector. Since $\partial_\mu (T^{\mu\nu} v_\nu) = (\partial_\mu T^{\mu\nu}) v_\nu = 0$ we can can use the divergence theorem to prove that, if $T^{\mu\nu}$ decays suffuciently rapidly at spacelike infinity (*),
$$\int_\Sigma T^{\mu\nu}v_\nu n^{(\Sigma)}_\mu d^3x = \int_{\Sigma'} T^{\mu\nu}v_\nu n^{(\Sigma')}_\mu d^3x\tag{1} $$
for every choice of spatial rest spaces $\Sigma, \Sigma'$ of (generally different) Minkowski reference frames, where $n^{(\Sigma)}$ denotes the unit normal vector to $\Sigma$.
Eq.(1) proves that the linear map $$ V^4\ni v \mapsto \int_\Sigma T^{\mu\nu}v_\nu n^{(\Sigma)}_\mu d^3x $$
is well defined,
where $V^4$ is the vector space of translations in $M^4$. This map does not depend on the choice of $\Sigma$. Every linear map on contravariant vectors defines a covariant vector. Therefore, there is a well defined $P\in (V^4)^*$ such that
$$\langle P, v\rangle = P_\nu v^\nu = \int_\Sigma T^{\mu\nu}v_\nu n^{(\Sigma)}_\mu d^3x \:.$$
In other words,
$$P_\nu := \int_\Sigma T^{\mu\nu} n^{(\Sigma)}_\mu d^3x$$
is a well defined covariant four vector, the total four momentum of the system, independently from the choice of $\Sigma$.
This procedure, in Minkowski spacetime, is general: if you have a tensor field $T^{\mu\mu_1\cdots \mu_n}(x)$ of order $n+1$ satisfying a conservation rule
$$\partial_\mu T^{\mu\mu_2\cdots \mu_n}=0\:,$$
then, provided it decreases sufficiently rapidly at spatial infinity,
$$Z^{\mu_1\cdots \mu_n} := \int_\Sigma T^{\mu\mu_1\ldots \mu_n} n^{(\Sigma)}_\mu d^3x$$
is a well defined tensor of order $n$.
Notice that we pass from tensor fields to tensors.
The crucial step is the fact that it makes sense to sum (integrate) tensors defined at different events on $\Sigma$.
It makes sense just because we are dealing with an affine space, where the spaces of tensors $(V^4)^{n\otimes}$ are well defined abstractly, without referring to events where they are applied. All this procedure cannot be used in curved spacetime (except for the case of a vector field, a current $J^\mu$, which gives rise to a scalar $Q$.).
Also sticking to Minkowski spacetime, it is clear that the procedure defined tensors and not tensor fields. Since they are not applied to a specific event it does not make sense to discuss if they are tensors with respect to general changes of coordinates, since the transformation matrix would depend on the place.
Another point is the relevance of the conservation law. If the conservation law did not work, in principle, we could not have a well defined integral, since it could depend on the chosen spatial slice.
However, I am not sure that independence from the chosen flat integration slice is equivalent to a local conservation law.
(It is worth stressing that the spacelike Cauchy surface $\Sigma$ is allowed to be curved also in Minkowski spacetime without affecting the above result.)
(*) The proof of this fact is actually much more difficult than one expects at first glance. A rigorous proof of it can be readapted from the one of Proposition 37 in this paper recently published in Lett Math Phys by me and C. De Rosa.}
