  We study the non-equilibrium dynamics of the zero temperature Mott insulator-
superfluid quantum phase transition in a lattice of weakly coupled
Bose-Einstein condensates. We show that crossing the critical point from the
insulating to the superfluid phase at a finite rate results in a finite
dispersion of current. Viceversa, crossing the critical point from the
superfluid phase to the insulating one, results in a finite dispersion of the
number of atoms per lattice site. We develop a microscopic dynamical model of
the transition, and make quantitative predictions for realistic experiments.
