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) $\mathrm{Supp}({\mathrm{Ext}}_R^t(L;N))\subseteq {\cup }_{i=0}^t\mathrm{Supp}({\mathrm{Ext}}_R^i(M;N))$;

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

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 ${\cup }_{n\in \mathbb{N}}\mathrm{Ass}({\mathrm{Ext}}_R^t(M/{\mathfrak{a}}^{n}M,N))$ is finite. In particular, if $\mathrm{grade}(\mathfrak{a},N)⩾t$ then the set  $\mathrm{Ass}({\mathrm{Ext}}_R^t(M/{\mathfrak{a}}^{n}M,N))$ is finite.