  Non-thermal dynamical critical behavior can arise in isolated quantum systems
brought out of equilibirum by a change in time of their parameters. While this
phenomenon has been studied in a variety of systems in the case of a sudden
quench, here we consider its sensitivity to a change of protocol by considering
the experimentally relevant case of a linear ramp in time. Focusing on the
$O(N)$ model in the large $N$ limit, we show that a dynamical phase transition
is always present for all ramp durations and discuss the resulting crossover
between the sudden quench transition and one dominated by the equilibrium
quantum critical point. We show that the critical behavior of the statistics of
the excitations, signaling the non-thermal nature of the transition are robust
against changing protocol. An intriguing crossover in the equal time
correlation function, related to an anomalous coarsening is also discussed.
