I am currently trying to understand intermediate extensions of perverse sheaves, specifically the proof of Gabber's purity theorem, which states that the intermediate extension of a pure perverse sheaf is pure.

As part of the proof, in §5 of "Faisceuax pervers" by Bernstein, Beilinson & Deligne, I've come across the following claim, which is not elaborated on in the paper and which I am struggling to understand.

Let $k$ be a finite field, and suppose that I have an open immersion $j:U\rightarrow X$ of $k$-varieties, and a 'projection' $f:X\rightarrow \mathbb{A}^1_k$. Fix a perverse $\overline{\mathbb{Q}}_\ell$ sheaf $\mathcal{F}$ on $U$. Then the claim is that for almost all closed points $v\in \mathbb{A}^1_k$, taking intermediate extensions of $\mathcal{F}$ commutes with pulling back to the fibre over $v$. In other words, for almost all $v$, looking at the commutative diagram

$f^{-1}(v)\cap U \overset{i}{\rightarrow} U$

$\begin{matrix} &&\downarrow j && \downarrow j \end{matrix}$

$\begin{matrix}f^{-1}(v)&\overset{i}{\rightarrow} &X\end{matrix}$

then $i^*\mathcal{F}[-1]$ and $i^*(j_{!*}\mathcal{F})[-1]$ are both perverse, and $j_{!*}(i^*\mathcal{F}[-1])=i^*(j_{!*}\mathcal{F})[-1]$.

Why is this true?

I also have another closely related question, which comes up in trying to understand Delinge's proof of Weil II in terms of perverse sheaves. Suppose that $Y$ is smooth and connected, and that I have a lisse $\overline{\mathbb{Q}}_\ell$-sheaf $\mathcal{F}$ on some relative curve $X\rightarrow Y$, which admits a good compactification $j:X\hookrightarrow \overline{X}$ into a smooth and proper curve $\overline{X}$ over $Y$ whose complement is finite étale over $Y$. Then does taking intermediate extensions of $\mathcal{F}$ commute with pulling back to closed points of $y$? In other words, if I have a closed point $y\in Y$ then should I expect to have something like $(j_{!*}\mathcal{F}[-\dim X])_y \cong j_{!+}(\mathcal{F}_y[-\dim X_y])$ as perverse sheaves on $\overline{X}_y$? Does this basically follow from the first question, or at least from its method of proof, by repeatedly cutting $Y$ with divisors?

Any help with either of these two questions would be greatly appreciated!