This question has been inspired by an answer to the question Reference request: Topology on the space of smooth compact submanifolds; I've asked it in a comment to that answer but then decided to make it a separate question, as it is also related to one of my previous questions, "Naïve"cobordism?.

The question I refer to is about the space of $m$-dimensional submanifolds of $\mathbb R^n$, and the above answer describes it as the disjoint sum of spaces of the form $\mathrm{Emb}(M,\mathbb R^n) /\mathrm{Diff}(M)$. In particular, each diffeomorphism type of $m$-manifolds produces a separate connected component of this space.

My question is whether any kind of compactification of this space is known which would tie together all these components.

Two things that come to mind in connection to this are

(a) the Deligne-Mumford compactification of the moduli spaces of curves via stable curves;

(b) Vassiliev invariants which are constructed via adding to the space of smooth 1-submanifolds of $\mathbb R^3$ new points corresponding to certain non-embedding immersions.

Is there actually a relationship between these two? Are there any higher dimensional analogs of these constructions known?

My personal motivation: a map from a (say, connected) space $X$ to the above disjoint sum should give a fibration over $X$ with fibre an $m$-manifold (plus some additional structure describing this fibre as a submanifold of $\mathbb R^n$); whereas a map to that purported compactification would instead give a family over $X$ which "mostly" consists of such manifolds but allows for "jumping" between different diffeomorphism types through (more or less) controllable "mild" singularities. This would give a picture which I was asking about in my above question about "naïve" cobordism.