Let $p \in \mathbb{Z}$ be a prime. Suppose we have a quasi-finite map $f \colon X \to \mathbb{A}^n$ of schemes over $\mathbb{Z}$, and let $f_p \colon X_{\mathbb{F}_p} \to \mathbb{A}_{\mathbb{F}_p}^n$ be the basechange of $f$ along $\operatorname{Spec} \mathbb{F}_p \to \operatorname{Spec} \mathbb{Z}$. Let $H \subset \mathbb{A}_{\mathbb{F}_p}^n$ be a hypersurface defined over $\mathbb{F}_p$, and let $U = \mathbb{A}_{\mathbb{F}_p}^n \setminus H$ be the complement. Suppose that neither $U(\mathbb{F}_p)$ nor $H(\mathbb{F}_p)$ is empty.
Question: Suppose that there is some number $N$ such that the fiber cardinality $\# f_p^{-1}(x) = N$ for every $x \in U$. Can anything be said about the fiber cardinality for $x \in H$? Is it true, for example, that $f_p^{-1}(x) \geq N$ for every $x \in H$?
What I know: I'm aware of the standard results on upper-semicontinuity of fiber cardinality / dimension over algebraically closed fields, so I was wondering what can be said over finite fields.
As $f$ is quasi finite it is locally of finite type (EGA II 6.2.2 and 6.2.3) and this ensures (EGA IV 13.1.3, Chevalley's theorem) that $x \mapsto {\textrm{dim}}_x(f^{-1}\left(f(x)\right))$ is upper semi-continuous.