The relative Breuil-Kisin classification of $p$-divisible groups and finite flat group schemes

Assume that $p>2$, and let $\mathscr{O}_K$ be a $p$-adic discrete valuation ring with residue field admitting a finite $p$-basis, and let $R$ be a formally smooth formally finite-type $\mathscr{O}_K$-algebra. (Indeed, we allow slightly more general rings $R$.) We construct an anti-equivalence of categories between the categories of $p$-divisible groups over $R$ and certain semi-linear algebra objects which generalise $(\varphi,\mathfrak{S})$-modules of height $\leqslant1$ (or Kisin modules). A similar classification result for $p$-power order finite flat group schemes is deduced from the classification of $p$-divisible groups. We also show compatibility of various construction of ($\mathbb{Z}_p$-lattice or torsion) Galois representations, including the relative version of Faltings' integral comparison theorem for $p$-divisible groups. We obtain partial results when $p=2$.