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 M. For a homogeneous ideal I, and for each fixed k N, we show the asymptotic linearity of v-numbers of the graded modules ExtR^k (L, I^{nM}/I^{nN}) and Torₖ^R (L, I^{nM}/I^{nN}) as functions of n. Moreover, under some conditions on ExtRᵏ (L, M) and TorₖR (L, M) respectively, we prove similar behaviour for v-numbers of ExtR^k (L, M/I^{nN}) and Torₖ^R (L, M/I^{nN}). The last result is obtained by proving the asymptotic linearity of v-number of (U+I^nV) /I^nW, where U, V and W are graded submodules of a finitely generated graded R-module such that W V and (0: ₔI) = 0.