Looking at families of abelian varieties over a base curve, relations have been established between the Néron-Tate height in the fibers along a section and a height in the base following work of Tate, Silverman, Call, Lang and Green in the 1980s. It is natural to wonder if the same properties carry over to the more general setting of families of rational maps, equipped with the Call-Silverman canonical height, even in the simplest case of maps on P^1. A first answer to this question is given by the `specialization result’ of Call and Silverman from 1993. I will discuss improvements to this result. This is joint work with Laura DeMarco.