Vanishing of $\text{Ext}^i_{X}(E, F)$ vs. $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ for $E,F \in \mathcal D^b(\text{Coh } X)$

Question

Let $X$ be a Noetherian scheme and $E,F \in \mathcal D^b(\text{Coh } X)$. Let $x\in X$ be a closed point. Then, is there any connection between $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ and $\text{Ext}^i_{X}(E, F)$ ? In particular, does vanishing of $\text{Ext}^i_{X}(E, F)$ for all $i\gg 0$ imply vanishing of $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ for all $i\gg 0$ ?