  We provide a rigorous derivation of Gutzwiller mean-field dynamics for
lattice bosons, showing that it is exact on fully connected lattices. We apply
this formalism to quenches in the interaction parameter from the Mott insulator
to the superfluid state. Although within mean-field the Mott insulator is a
steady state, we show that a dynamical critical interaction $U_d$ exists, such
that for final interaction parameter $U_f>U_d$ the Mott insulator is
exponentially unstable towards emerging long-range superfluid order, whereas
for $U_f<U_d$ the Mott insulating state is stable. We discuss the implications
of this prediction for finite-dimensional systems.
