We start by showing that for any 1-parameter family of genus g>2 curves, the number of rational points is bounded by g, degree of the field, and the Mordell-Weil rank. Apart from the classical Faltings-Vojta-Bombieri method, the new input is a height inequality recently proved (joint with Philipp Habegger). Then I'll explain some generalization of this method to an arbitrary family of curves. I'll focus on how the mixed Ax-Schanuel for universal abelian varieties, extension of a recent result of Mok-Pila-Tsimerman, applies to this problem. This is work in progress, joint with Vesselin Dimitrov and Philipp Habegger.