  A system gradually driven through a symmetry-breaking phase transition is
subject to the Kibble-Zurek mechanism (KZM). As a consequence of the critical
slowing down, its state cannot follow local equilibrium, and its evolution
becomes non-adiabatic near the critical point. In the simplest approximation,
that stage can be regarded as "impulse" where the state of the system remains
unchanged. It leads to the correct KZM scaling laws. However, such "freeze-out"
might suggest that the coherence length of the nascent order parameter remains
unchanged as the critical region is traversed. By contrast, the original
causality-based discussion emphasized the role of the {\it sonic horizon}:
domains of the broken symmetry phase can expand with a velocity limited by the
speed of the relevant sound. This effect was demonstrated in the quantum Ising
chain where the dynamical exponent $z=1$ and quasiparticles excited by the
transition have a fixed speed of sound. To elucidate the role of the sonic
horizon, in this paper we study two systems with $z>1$ where the speed of sound
is no longer fixed, and the fastest excited quasiparticles set the size of the
sonic horizon. Their effective speed decays with the increasing transition
time. In the extreme case, the dynamical exponent $z$ can diverge such as in
the Griffiths region of the random Ising chain where localization of excited
quasiparticles freezes the growth of the correlation range when the critical
region is traversed. Of particular interest is an example with $z<1$ -- the
long-range extended Ising chain, where there is no upper limit to the velocity
of excited quasiparticles with small momenta. Initially, the power-law tail of
the correlation function grows adiabatically, but in the non-adiabatic stage it
lags behind the adiabatic evolution -- in accord with a generalized
Lieb-Robinson bound.
