  We prove that Zhang's dynamical Bogomolov conjecture holds uniformly along
$1$-parameter families of rational split maps and curves. This provides
dynamical analogues of recent results of Dimitrov-Gao-Habegger and K\"uhne. In
fact, we prove a stronger Bogomolov-type result valid for families of split
maps in the spirit of the relative Bogomolov conjecture. We thus provide first
instances of a generalization of a conjecture by Baker and DeMarco to higher
dimensions. Our proof contains both arithmetic and analytic ingredients. We
establish a characterization of curves that are preperiodic under the action of
a non-exceptional split rational endomorphism $(f,g)$ of
$(\mathbb{P}^1_{\mathbb{C}})^2$ with respect to the measures of maximal entropy
of $f$ and $g$, extending a previous result of Levin-Przytycki. We further
establish a height inequality for families of split maps and varieties
comparing the values of a fiber-wise Call-Silverman canonical height with a
height on the base and valid for most points of a non-preperiodic variety. This
provides a dynamical generalization of a result by Habegger and generalizes
results of Call-Silverman and Baker to higher dimensions. In particular, we
establish a geometric Bogomolov theorem for split rational maps and varieties
of arbitrary dimension.
