  We study the convergence of earthquake paths and horocycle paths in the
Gardiner-Masur compactification of Teichm\"uller space. We show that an
earthquake path directed by a uniquely ergodic or simple closed measured
geodesic lamination converges to the Gardiner-Masur boundary. Using the
embedding of flat metrics into the space of geodesic currents, we prove that a
horocycle path in Teichm\"uller space, induced by a quadratic differential
whose vertical measured foliation is unique ergodic, converges to the
Gardiner-Masur boundary and to the Thurston boundary.
