Equidistribution of small points over finitely generated fields

We study the equidistribution of small points over finitely generated fields, in connection with a conjecture of Yuan–Zhang. We first give a counterexample showing that numerical smallness, defined using Moriwaki heights for all polarizations, does not imply equidistribution at every valuation. We then prove that numerically small points do equidistribute at every fully transcendental valuation. In particular, when $F$ is the function field of a curve $B\mathrm{/}\mathbb{Q}$, these valuations correspond to points of types $2$, $3$, and $4$ in the Berkovich analytification of $B_{{\mathbb{Q}}_p}$.