  In a previous paper, the authors extended Mirzakhani's (almost-everywhere
defined) measurable conjugacy between the earthquake and horocycle flows to a
measurable bijection. In this one, we analyze the continuity properties of this
map and its inverse, proving that both are continuous at many points and in
many directions. This lets us transfer measure convergence between the two
systems, allowing us to pull back results from Teichm{\"u}ller dynamics to
deduce analogous statements for the earthquake flow.
