  We study the effects of symmetry-breaking perturbations in the
out-of-equilibrium quantum dynamics of many-body systems slowly driven across a
continuous quantum transition (CQT) by a time-dependent symmetry-preserving
parameter. For this purpose, we analize the out-of-equilibrium dynamics arising
from generalized Kibble-Zurek (KZ) protocols, within a dynamic
renormalization-group framework allowing for finite-size systems. We show that
the time dependence of generic observables develops an out-of-equilibrium
finite-size scaling (FSS) behavior, arising from the interplay between the time
scale $t_s$ of the parameter variations in the KZ protocol, the size $L$ of the
system, and the strength $h$ of the symmetry-breaking perturbation, in the
limit of large $t_s$ and $L$. Moreover, scaling arguments based on the
first-order adiabatic approximation of slow variations in quantum systems allow
us to characterize the approach to the adiabatic regimes for some limits of the
model parameters (for example when we take $t_s\to \infty$ before
$L\to\infty$), predicting asymptotic power-law suppressions of the nonadiabatic
behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by
numerical analyses for the paradigmatic quantum Ising chain along generalized
KZ protocols, with a time-dependent transverse field crossing its CQT, in the
presence of a static longitudinal field breaking its $Z_2$ symmetry.
