Ok so I'm struggling with the concept of conservation in the divergence theorem.
Divergence theorem states that: $$ \iiint_O {\nabla \cdot {\bf F}}\ dV = \iint_{S=\partial O} ({\bf F}\cdot \hat{{\bf n}}) d{S}$$
So, let's say your vector field had units of 1/length^2 (essentially a flux) - this means both quantities (right and left hand side of the equality) should give you the net total of some quantity (let's call it mass) passing through the boundary surface. In the case of isotropically outward emitting vector field (centered at the origin), I wouldn't expect, conceptually, that either of these integrals would change as you move outward in $r$ on a sphere (because your overall mass must be conserved).
That is, in the case I described, ${\bf F}=\hat{{\bf n}}$ and $\hat{{\bf n}}=x \hat{i}+y\hat {j}+z\hat{k}$ thus $\nabla \cdot {\bf F} = 1$ and $ \iiint_O {\nabla \cdot {\bf F}}\ dV = V$