  We propose a new moduli-theoretic approach to the $p$-adic Simpson
correspondence for a smooth proper rigid space $X$ over $\mathbb C_p$ with
coefficients in any rigid analytic group $G$, in terms of a comparison of
moduli stacks. For its formulation, we introduce the class of "smoothoid
spaces" which are perfectoid families of smooth rigid spaces, well-suited for
studying relative $p$-adic Hodge theory. For any smoothoid space $Y$, we then
construct a "sheafified non-abelian Hodge correspondence", namely a canonical
isomorphism \[R^1\nu_{\ast}G\xrightarrow{\sim} \mathrm{Higgs}_G\] where
$\nu:Y_{v}\to Y_{et}$ is the natural morphism of sites, and where
$\mathrm{Higgs}_G$ is the sheaf of isomorphism classes of $G$-Higgs bundles on
$Y_{et}$. We also prove a generalisation of Faltings' local $p$-adic Simpson
correspondence to $G$-bundles and to perfectoid families.
  We apply these results to deduce $v$-descent criteria for \'etale $G$-bundles
which show that $G$-Higgs bundles on $X$ form a small $v$-stack $\mathscr
Higgs_G$. As a second application, we construct an analogue of the Hitchin
morphism on the Betti side: a morphism $\mathscr Bun_{G,v}\to \mathcal A_G$
from the small $v$-stack of $v$-topological $G$-bundles on $X$ to the Hitchin
base. This allows us to give a conjectural reformulation of the $p$-adic
Simpson correspondence for $X$ in a more geometric and more canonical way,
namely in terms of a comparison of Hitchin morphisms.
