For an Abelian scheme over a ring of integers in a number field, Faltings has a theorem that describes how the Faltings' height changes through an isogeny. There are multiple references for this statement and proof. See Proposition 4.1 in the notes Faltings height by Daniele Agostini (pdf), for instance.

I am interested in the analogous statement over curves over a finite field, in particular when the isogeny has degree a power of $p$, the characteristic of the finite field.

Is any such statement true in this case and if so, what's a reference?