Is there any kind of generalization of Vélu formulae for Jacobians?

The question technically is:

Given a hyperelliptic curve $H/\overline{\mathbb{F}}_q$ and its jacobian $J_H$ of dimension $2$, if $D\in J_H$:

Is there a way to construct an isogeny $\psi:J\to{J_D}$ with kernel $nD$ for all $n\in \mathbb{Z}$ and construct the Jacobian variety $J_D$ of some other hyperelliptic curve? (maybe $J_D$ should be an abelian variety in general)

Is well known that elliptic curve isogenies have specific form, this helps to represent easily an isogeny.

What is known about isogenies between genus 2 jacobians for example?

I know that if $J$ is non simple, this could make things easier, and maybe reduce it to Vélu formulae in an $(a,b)$-isogeny, but, if $J$ is simple?

Is it possible to use the Affine part of a Jacobian of dimension 2 to reconstruct a hyperelliptic curve of genus 2 using an isogeny in Mumford representation?