  This survey is a continuation of the study undertaken in \cite{AS18}. We
examine the local structure of Bridgeland moduli spaces $M_\sigma(v,\D)$, where
the relevant triangulated category $\D$ is either the bounded derived category
$\D=\D^b(X)$ of a K3 surface $X$, or the Kuznetsov component $\D=\Ku(Y)\subset
\D ^b(Y)$ of a smooth cubic fourfold $Y\subset \PP^5$. For these moduli spaces,
building on \cite{Bmm19}, \cite{Bmm21} we give a direct proof of formality and,
using their local isomorphism with quiver varieties, we establish their
normality and their irreducibility, as long as $\sigma$ does not lie on a
totally semistable wall. We then connect the variation of GIT quotients for
quiver varieties with the changing of stability conditions on moduli spaces.
