Asymptotic behavior of homological invariants of localizations of modules

Let $R$ be a commutative noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of $M\mathrm{/}{I}^{n}M$ at prime ideals of $R$ and prove that these invariants are stable or have polynomial growth for large integers $n$ that do not depend on the prime ideals.