  We prove a rigidity result for automorphisms of points of certain stacks
admitting adequate moduli spaces. It encompasses as special cases variations of
the moduli of $G$-bundles on a smooth projective curve for a reductive
algebraic group $G$. For example, our result applies to the stack of semistable
$G$-bundles, stacks of semistable Hitchin pairs, and stacks of semistable
parabolic $G$-bundles. Similar arguments apply to Gieseker semistable
$G$-bundles in higher dimensions. We present two applications of the main
result. First, we show that in characteristic $0$ every stack of semistable
decorated $G$-bundles admitting a quasiprojective good moduli space can be
written naturally as a $G$-linearized global quotient $Y/G$, so the moduli
problem can be interpreted as a GIT problem. Secondly, we give a proof that the
stack of semistable meromorphic $G$-Higgs bundles on a family of curves is
smooth over any base in characteristic $0$.
