Let us consider a complete Riemannian manifold $M$ of dimension $n$ with $Ric \geq 0$. Then the Bishop-Gromov volume comparison theorem says that for any $p \in M$, the function $$ \frac{\text{Vol}(B(p, r))}{r^n}$$ is monotonically decreasing on $r \in (0, \infty)$. My question is, how global is this result?

More concretely, let's say that we perturb $M$ so that $M \setminus K$ now has $Ric \geq 0$, where $K$ is compact. Does the same conclusion hold here? To be more specific, consider the case where $p \in K$. Then do we still have that $$ \frac{\text{Vol}(B(p, r))}{r^n}$$ is monotonically decreasing for large enough $r$? Thanks in advance for any insight!