Isogeny theorems are a powerful number-theoretic tool to understand subgroup of elliptic curves over number fields and through those, properties of their rational points. As a prerequisite for such theorems, one needs to understand how the "height" h(E) of an elliptic curve E can change by an isogeny: if E and E' are elliptic curves over a number field K with an isogeny phi : E -> E' over K, the difference |h(E)-h(E')| is linearly bounded in terms of log(deg(phi)). In a recent paper, Griffon and Pazuki proved a similar result for elliptic curves over function fields of curves, which is surprisingly much more uniform on the degree of phi (for example, in characteristic 0 the height is invariant by isogeny !). In this talk, I will recall what are abelian varieties and isogenies between them and what are their heights, why Griffon-Pazuki's result is not as easily generalised as it seems (abelian varieties being the higher-dimensional avatars of elliptic curves) because of group schemes in characteristic p, and describe the optimal bounds we obtained with Griffon and Pazuki in the context of function fields