Are equally weighted MSTs closely related?
Yes. More generally, any spanning tree can be reached from any arbitrarily-chosen MST by a sequence of single-edge replacement operations that never decrease the weight. When such a sequence is used to reach another MST, it can also never increase the weight, because then it would have to decrease it at some other step. See e.g.

H.N. Gabow, Two Algorithms for Generating Weighted Spanning Trees in Order, SIAM J. Comput. 6, 1977, 139–150. http://epubs.siam.org/doi/pdf/10.1137/0206011