  Quantum quenches in continuum field theory across critical points are known
to display different scaling behaviours in different regimes of the quench
rate. We extend these results to integrable lattice models such as the
transverse field Ising model on a one-dimensional chain and the Kitaev model on
a two-dimensional honeycomb lattice using a nonlinear quench protocol which
allows for exact analytical solutions of the dynamics. Our quench protocol
starts with a finite mass gap at early times and crosses a critical point or a
critical region, and we study the behaviour of one point functions of the
quenched operator at the critical point or in the critical region as a function
of the quench rate. For quench rates slow compared to the initial mass gap, we
find the expected Kibble-Zurek scaling. In contrast, for rates fast compared to
the mass gap, but slow compared to the inverse lattice spacing, we find scaling
behaviour similar to smooth fast continuum quenches. For quench rates of the
same order of the lattice scale, the one point function saturates as a function
of the rate, approaching the results of an abrupt quench. The presence of an
extended critical surface in the Kitaev model leads to a variety of scaling
exponents depending on the starting point and on the time where the operator is
measured. We discuss the role of the amplitude of the quench in determining the
extent of the slow (Kibble-Zurek) and fast quench regimes, and the onset of the
saturation.
