Reedy Model Structures in Families

Given a family of model categories $\mathcal{E}\to \mathcal{R}$ over a Reedy category, we outline a set of conditions which lead to the existence of a Reedy model structure on the category of sections $\mathsf{S}\mathsf{e}\mathsf{c}\mathsf{t}(\mathcal{R},\mathcal{E})$. We prove that for a wide class of examples, this model structure serves as a strictification of the $(\mathrm{\infty },1)$-category of sections of the higher-categorical family associated to $\mathcal{E}\to \mathcal{R}$.