  Let $G$ be a reductive group scheme over the $p$-adic integers, and let $\mu$
be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a
functor from nilpotent $(G,\mu)$-displays over $p$-nilpotent rings $R$ to
formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors.
When $R/pR$ has a $p$-basis \'etale locally, we show that this defines an
equivalence between the two categories. The definition of the functor relies on
the construction of a $G$-crystal associated with any adjoint nilpotent
$(G,\mu)$-display, which extends the construction of the Dieudonn\'e crystal
associated with a nilpotent Zink display. As an application, we obtain an
explicit comparison between the Rapoport-Zink functors of Hodge type defined by
Kim and by B\"ultel and Pappas.
