By the book, Degeneration of abelian varieties-[Faltings G , Chai C ],
a semi-abelian scheme is a smooth separated commutative group scheme $\pi : G\rightarrow S$ with geometrically connected fibres, such that each fibre $G_s$ is an extension of an abelian variety $A_s$ by a torus $T_s$: $0\rightarrow T_s \rightarrow G_s\rightarrow A_s\rightarrow 0$.
Then what's the canonical definition of an isogeny between semi-abelian schemes over $S$? And what's the reference?
I am thinking about candidates like
"evey fibre morphism $f_s:A_s\rightarrow B_s $ where $s\in S$ is an isogeny of abelian varieties over Spec $\kappa(s)$"
or
"finite and surjective $S$-morphism"
The problem is I want flatness just like ordinary isogenies between abelian varities over a field, I don't know if any of my candidates of definition imply flatness.
There's a definition in the book "Néron Models" of Bosch/Luetkebohmert/Raynaud on page 180: Let $f:G\to G'$ be a homomorphism of commutative group schemes of finite type over an arbitrary base scheme $S$. Then $f$ is called an isogeny if, for each $s\in S$, the homomorphism $f_s:G_s\to G'_s$ is an isogeny in the classical sense, i.e., if $f_s$ is finite and surjective on identity components.
In your case the fibers are semi-abelian varieties so the identity component of $G_s$ is $G_s$ itself. So your first guess is just (a special case of) the definition in Bosch/Luetkebohmert/Raynaud.
I'm not sure how "canonical" this definition is. It's the only reference I know.
You can test flatness fiber wise.