Kisin modules with descent data and parahoric local models

We construct a moduli space $Y^{\mu, \tau}$ of Kisin modules with tame descent datum $\tau$ and with fixed $p$-adic Hodge type $\leq \mu$, for some finite extension $K/\mathbb{Q}_p$. We show that this space is smoothly equivalent to the local model for $\mathrm{Res}_{K/\mathbb{Q}_p} \mathrm{GL}_n$, cocharacter $\{ \mu \}$, and parahoric level structure. We use this to construct the analogue of Kottwitz-Rapoport strata on the special fiber $Y^{\mu, \tau}$ indexed by the $\mu$-admissible set. We also relate $Y^{\mu, \tau}$ to potentially crystalline Galois deformation rings.