Quasicoherent sheaves over stack quotient of affine $k$-scheme by affine group over $k$ are equivariant modules?

These days I've been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on infinite root stacks. At the moment I'm looking at their proposition 4.9. $\def\Spec{\operatorname{Spec}}\def\Aff{\operatorname{Aff}}\def\QCoh{\operatorname{QCoh}}\def\pr{\operatorname{pr}}$

Proposition 4.9 If $X=\Spec A$ is an affine scheme and $G$ is an affine group scheme over $\mathbb Z$ acting on $X$, then there is an equivalence of tensor categories between the quasi-coherent sheaves over the fpqc quotient $[X / G]$ and the $G$-equivariant $A$-modules.

I would like to think that this is also true relative to any commutative ring $k$, say, for an affine $k$-scheme $X=\Spec A$ (i.e. $A$ is a $k$-algebra) and an affine group scheme $G$ over $k$ acting on $X$, then I would like to prove that the quasi-coherent sheaves over the fpqc quotient $[X / G]$ (which is now a stack over $(\Aff / k)$) are just the $G$-equivariant $A$-modules. How can I prove this?

At first I wanted to use 4.9 to prove the relative version I want. In the original statement, the quotient $[X / G]$ is taken in the category of stacks over $(\Aff)$, while in the desired generalization, $[X / G]$ is a quotient in the category of stacks over $(\Aff / k)$, so I was thinking about this: write $f$ for the forgetful functor from fibred categories over $(\Aff / k)$ to fibred categories over $(\Aff)$. In the first place, I don't think $f[X / G]$ is a stack over $(\Aff)$, and even if it were, we wouldn't have $f[X / G] \simeq [f(X) / f(G)]$, right? But it we had something of this sort, then we could simply apply the same proposition 4.9 and we would be done. But I don't have much hope in this direction.

Then I remembered that the authors deduced proposition 4.9 from a previous proposition 4.6:

Proposition 4.6 Let $\mathscr X$ be a fibred category over $(\Aff)$ with an fpqc atlas $U\to \mathscr X$. Then the pullback $R=U\times_{\mathscr X}U$ is represented by a scheme, and we obtain an fpqc groupoid $R\rightrightarrows U$. Write $\QCoh(R\rightrightarrows U)$ for the category of quasicoherent sheaves $F$ on $U$ equipped with descent data $\pr_1^*F\simeq\pr_2^*F$. Then there is an additive equivalence between $\QCoh\mathscr X$ and $\QCoh(R\rightrightarrows U)$.

so I thought if I understood how prop. 4.9 follows from prop 4.6, I could try to re-prove prop 4.9 but now in the relative case I want. However, I don't quite understand how is prop. 4.6 used to deduce 4.9. If $X\to [X / G]$ is an fpqc cover, and $R$ is the self-pullback of this map, then how is $\QCoh(R\rightrightarrows X)$ the category of $G$-equivariant $A$-modules? I mean, yeah, a quasi-coherent sheaf $F$ over $X$ is just an $A$-module, but how is the descent data $\pr_1^*F\simeq\pr_2^*F$ giving it a $G$-equivariant structure?

Furthermore, is this strategy plausible? i.e. can one deduce the relative version from prop 4.6?

Thanks in advance.