For an Abelian scheme over a ring of integers in a number field, Falting's has a theorem that describes how the Falting's height changes through an isogeny. There are multiple references for this statement and proof. See page 7 of this, 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?