  The dynamical quantum phase transitions (DQPTs) in quantum spin chains with
gapless phases after a sudden quench are studied. We mainly consider the
general systems with asymmetrical quasiparticle excitation spectra and obtain
the general expression of the Loschmidt echo as well as the general conditions
for the occurrence of DQPTs. As two examples, we study the DQPTs in the
\emph{XY} chains with Dzyaloshinskii-Moriya interaction and \emph{XZY-YZX} type
of three-site interaction. It's found that the DQPTs may not occur in the
quench across the quantum phase transitions regardless of whether the quench is
from the gapless phase to gapped phase or from the gapped phase to gapless
phase. This is different from the DQPTs in the case of quench from the gapped
phase to gapped phase, in which the DQPTs will always appear. Besides, we also
analyze the different reasons for the absence of DQPTs in the quench from the
gapless phase and the gapped phase.
