We will present a new notion of isogeny between '$p$-divisible groups with additional structure' that employs the cohomological stacks of Drinfeld and Bhatt-Lurie—-in particular the theory of apertures developed in prior work with Gardner—-and combines it with some invariant theoretic tools familiar to the geometric Langlands and representation theory community, namely the Vinberg monoid and the wonderful compactification. This gives a uniform construction of $p$-Hecke correspondeces and Rapoport-Zink spaces associated with unramified local Shimura data. In particular, we give the first general construction of $RZ$ spaces associated with exceptional groups. This work is joint with Si Ying Lee.