I am looking for an example of an injective ring homomorphism $\phi\colon A\to B$ such that the following holds: there exists a prime ideal $\frak q$ of $B$ such that $\phi^{-1}(\frak q) = \frak p$, and the induced map $A_{\frak p} \to B_{\frak q}$ is not injective.
The algebro-geometric picture should be clear: if we put $X = \mathrm{Spec\, A}$ and $Y = \mathrm{Spec\, B}$, then $\phi$ induces a morphism $(\psi,\theta)\colon Y\to X$ of affine schemes with $\psi(\frak q) = \frak p$. The map $A_{\frak p} \to B_{\frak q}$ corresponds to the induced map on the level of stalks at $\frak q$ of the morphism of sheaves $\theta^\sharp \colon \psi^* \mathcal O_X \to \mathcal O_Y$ [I have used the notation from EGA I].