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Set $ A := Q/({\bf z}) $, where $ Q $ is a polynomial ring over a field, and$ {\bf z} = z_1,\ldots,z_c $ is a homogeneous $ Q $-regular sequence. Let $ M $and $ N $ be finitely generated graded $ A $-modules, and $ I $ be ahomogeneous ideal of $ A $. We show that  (1) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M, I^nN) \right) \le \rho_N(I)\cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b \mbox{ for all }i, n \ge 0 $,  (2) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M,N/I^nN) \right) \le \rho_N(I)\cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b' \mbox{ for all }i, n \ge 0 $,  where $ b $ and $ b' $ are some constants, $ f := \mathrm{min}\{\mathrm{deg}(z_j) : 1 \le j \le c \} $, and $ \rho_N(I) $ is an invariantdefined in terms of reduction ideals of $ I $ with respect to $ N $. There areexplicit examples which show that these inequalities are sharp.

An asymptotic bound for Castelnuovo-Mumford regularity of certain Ext modules over graded complete intersection rings