We work over an algebraically closed field $k$.
Let $G$ be a reductive group and $X$ be a smooth projective curve over $k$. It is proven in [1, Theorem 1.2] that the moduli of semi-stable principal $G$-bundles over $X$ of a given degree (or "topological type") is bounded.
It seems that the proof there could be readily generalized to principal $G$-bundles on a given family of curves. The precise statement should be:
Let $\pi: \mathcal X \to S$ be a family of smooth projective curves over a scheme $S$ of finite type over $k$. Fix a degree $d$ for principal $G$-bundles (c.f. [1, Definition 3.2]). Then there exists a scheme $T$ of finite type over $S$, together with a principal $G$-bundle $$ \mathcal P \to \mathcal X \times_{S} T, $$ such that for any closed point $s$ of $S$, the base-change of the above family to $s$ $$ \mathcal P_s \to \mathcal X_{s} \times T_{s}, $$ consists of all semi-stable $G$-bundles on $\mathcal X_s$ of degree $d$.
The proof in [1] reduces the boundedness of the moduli of semi-stable principal $G$-bundles to the boundedness of the moduli of line bundles of a fixed degree. Hence it seems that the proof should be generalizable without essential modification.
However, I could not find a direct reference in the literature. Does anyone know such a result in the literature?
[1] Holla, Yogish I.; Narasimhan, M. S., A generalisation of Nagata’s theorem on ruled surfaces., Compos. Math. 127, No. 3, 321-332 (2001). ZBL1047.14018.
