Asymptotic Castelnuovo-Mumford regularity of (co)homology modules involving powers of ideals

Let \(R\) be a Noetherian standard $\mathbb{N}$-graded ring, and let \(I_1,\ldots,I_r\) be homogeneous ideals of \(R\). Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded \(R\)-modules with \(N\subseteq M\). For \(\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r\), set \({\bf I}^{\underline{n