There are some properties that are easily studied for random d-regular graphs, but that are very hard to extend to random graphs with a given degree sequence (e.g. whether a graph is w.h.p. hamiltonian).

Does anyone have any other examples of papers/problems that successfully (and not trivially) extend properties from random regular graphs to random graphs with a given degree sequence?

(I hope this is not too open ended, I am finding it hard to search online the usual way because the key words are so generic).