  We study the quantum quench dynamics in an extended version of the Dicke
model where an additional parameter allows a smooth transition to the
integrable Tavis-Cummings regime. We focus on the influence of various quantum
phases and excited-state quantum phase transitions (ESQPTs) on the survival
probability of the initial state. We show that, depending on the quench
protocol, an ESQPT can either stabilize the initial state or, on the contrary,
speed up its decay to the equilibrated regime. Quantum chaos smears out the
manifestations of ESQPTs in quench dynamics, therefore significant effects can
only be observed in integrable or weakly chaotic settings. Similar features are
present also in the post-quench dynamics of some observables.
