Inspired by an analogy between torsion points in abelian varieties and preperiodic points of rational maps, Shou-Wu Zhang posed a dynamical generalization of the Manin--Mumford and Bogomolov conjectures. The goal of these conjectures, which remain largely open, is to classify varieties in $\mathbb{P}^n$ that contain a generic sequence of preperiodic points with respect to an endomorphism $f$ of $\mathbb{P}^n$, or more generally, points that are `small' with respect to the canonical height of $f$. In this talk we will discuss attempts to obtain uniform and quantitative results in the context of the dynamical Bogomolov conjecture. This is work in progress joint with Jit Wu Yap.