I will explain that many ``well-positioned'' subschemes of mixed characteristics models of Shimura varieties admit good (partial) toroidal and minimal compactifications, with familiar boundary stratifications and formal local structures, as if they were Shimura varieties in characteristic zero.  If time permits, I will also explain some generalizations of Koecher's principle and the relative vanishing of subcanonical extensions for coherent sheaves, and of Pink's and Morel's formulae for etale sheaves, to the context of such subschemes.  (This is joint work with Stroh.)