  We provide estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points
lying on fibers of flat morphisms between smooth varieties whose fibers have
rational singularities, termed (FRS) morphisms. For each individual fiber, the
estimates were known by work of Avni and Aizenbud, but we render them uniform
over all fibers. The proof technique for individual fibers is based on
Hironaka's resolution of singularities and Denef's formula but breaks down in
the uniform case. Instead, we use recent results from the theory of motivic
integration. Our estimates are moreover equivalent to the (FRS) property, just
like in the absolute case by Avni and Aizenbud.
