Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal

Let R be a Noetherian $N$-graded ring. Let L, M and N be finitely generated graded R-modules with $N\subseteq M$. For a homogeneous ideal I, and for each fixed $k\in N$, we show the asymptotic linearity of v-numbers of the graded modules ${\mathrm{Ext}}_R^k(L,I^{n}M/I^{n}N)$ and ${\mathrm{Tor}}_k^R(L,I^{n}M/I^{n}N)$ as functions of n. Moreover, under some conditions on ${\mathrm{Ext}}_R^k(L,M)$ and ${\mathrm{Tor}}_k^R(L,M)$ respectively, we prove similar behaviour for v-numbers of ${\mathrm{Ext}}_R^k(L,M/I^{n}N)$ and ${\mathrm{Tor}}_k^R(L,M/I^{n}N)$. The last result is obtained by proving the asymptotic linearity of v-number of $(U+I^{n}V)/I^{n}W$, where U, V and W are graded submodules of a finitely generated graded R-module such that $W\subseteq V$ and $\left(0{:}_{U}I\right)=0$.