Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, and $M, N$ two finitely generated $R$-modules. Let $t$ be a non-negative integer. It is shown that for any finitely generated $R$-module $L$ with $\mathrm{Supp}(L) \subseteq \mathrm{Supp}(M)$, the following statements hold:

(i) $\mathrm{Supp}(\mathrm{Ext}^t_R(L; N)) \subseteq \cup^t_{i=0} \mathrm{Supp}(\mathrm{Ext}^i_R(M; N))$;

(ii) $\mathrm{Ass}(\mathrm{Ext}^t_R(L; N)) \subseteq\mathrm{Ass}(\mathrm{Ext}^t_R(M; N)) \cup (\cup_{i=0}^{t-1}\mathrm{Supp}(\mathrm{Ext}^i_R(M; N)))$.

As an immediate consequence, we deduce that if $\mathrm{Supp}(H_{\mathfrak{a}}^i(N))$ or $\mathrm{Supp}(H_{\mathfrak{a}}^i (M; N))$ is finite for all $i < t$, then the set $\cup_{n\in\mathbb N} \mathrm{Ass}(\mathrm{Ext}^t_R(M/\mathfrak{a}^nM, N))$ is finite. In particular, if $\mathrm{grade}(\mathfrak{a}, N) \geqslant t$ then the set  $\mathrm{Ass}(\mathrm{Ext}^t_R(M/\mathfrak{a}^nM, N))$ is finite.