  For a smooth affine group scheme $G$ over the ring of $p$-adic integers and a
cocharacter $\mu$ of $G$, we develop the deformation theory for
$G$-$\mu$-displays over the prismatic site of Bhatt-Scholze, and discuss how
our deformation theory can be interpreted in terms of prismatic $F$-gauges
introduced by Drinfeld and Bhatt-Lurie. As an application, we prove the local
representability and the formal smoothness of integral local Shimura varieties
with hyperspecial level structure. We also revisit and extend some
classification results of $p$-divisible groups.
