  Let $k$ be an algebraically closed field of characteristic $p > 0$. We show
that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with
Hilbert--Kunz multiplicity less than $\lambda$ at all points $x\in X$, then for
a general hyperplane $H\subseteq\mathbb{P}^n_k$, the Hilbert--Kunz multiplicity
of $X\cap H$ is less than $\lambda$ at all points $x\in X\cap H$. This answers
a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker,
whose conclusion is the same as ours when $X\subseteq\mathbb{P}^n_k$ is normal.
In the process, we substantially generalize certain uniform estimates on
Hilbert--Kunz multiplicities of fibers of maps obtained by the aforementioned
authors that should be of independent interest.
