  We explore the robustness of universal dynamical scaling behavior in a
quantum system near criticality with respect to initialization in a large class
of states with finite energy. By focusing on a homogeneous XY quantum spin
chain in a transverse field, we characterize the non-equilibrium response under
adiabatic and sudden quench processes originating from a pure as well as a
mixed excited initial state, and involving either a regular quantum critical or
a multicritical point. We find that the critical exponents of the ground-state
quantum phase transition can be encoded in the dynamical scaling exponents
despite the finite energy of the initial state. In particular, we identify
conditions on the initial distribution of quasi-particle excitation which
ensure Kibble-Zurek scaling to persist. The emergence of effective thermal
equilibrium behavior following a sudden quench towards criticality is also
investigated, with focus on the long-time dynamics of the quasi-particle
excitation. For a quench to a regular quantum critical point, this observable
is found to behave thermally provided that the system is prepared at
sufficiently high temperature, whereas thermalization fails to occur in
quenches taking the system towards a multi-critical point. We argue that the
observed lack of thermalization originates in this case in the asymmetry of the
impulse region that is also responsible for anomalous multicritical dynamical
scaling.
