We know that the category of finite flat commutative affine group scheme over field is Abelian. Through out the proof, I think it can be generated to the case over the noertherian ring.
As in the page 33 of the lecture notes A course on finite flat group schemes and p-divisible groups of STIX, he states:
Theorem 20. Let $H$ be a finite flat closed normal subgroup of the affine algebraic $R$-group $G$. Then the quotient group sheaf $G / H$ is representable by an affine algebraic $R$-group, which is a categorical cokernel for the inclusion $H \subset G$.
The map $G \rightarrow G / H$ is finite and faithfully flat and $H \times{ }_R G \subset G \times{ }_R G$ is identical to $G \times{ }_{G / H} G$. Moreover, if $G$ is finite (resp. fpqc) over $R$, then $G / H$ is finite (resp. fpqc) over $R$.
So we can make quotient in the catogory.
And just as the proof in Pink's Finite group schemes in page 20, I don't see any obstruciton in the case of notherian ring.