In the literature, I've seen that people often conflate moduli spaces of manifolds (i.e. a space of submanifolds of a given manifold) with various classifying spaces of diffeomorphism groups. Could someone explain the usual procedure to go from some moduli space of manifolds to a classifying space, or vice versa.
In particular, I'm reading some of Randall-Wiliams' and Galatius' work, and at one point I read the statement "$\operatorname{BDiff(W, \partial W)}$, i.e. the moduli space of submanifolds...". If someone could explain even just this example, I'd appreciate it. This is found in the introduction of "Monoids of Moduli Spaces of Manifolds".