In the study of points of small canonical height on abelian varieties (or more generally, for polarized dynamical systems), one fundamental problem is to produce uniform and/or quantitative versions of key finiteness theorems on such points. In this talk, I will discuss a uniform quantitative Manin-Mumford theorem for higher genus curves defined over a one-dimensional function field. I will situate the proof of this theorem within the wider context of potential-theoretic approaches to questions of this style. This is joint work with Joe Silverman and Robert Wilms.