  We study the evolution of the holographic entanglement entropy (HEE) and the
holographic complexity (HC) after a thermal quench in $1+1$ dimensional
boundary CFTs dual to massive BTZ black holes. The study indicates how the
graviton mass $m_g$, the charge $q$, and also the size of the boundary region
$l$ determine the evolution of the HEE and HC. We find that for small $q$ and
$l$, the evolutions of the HEE and the HC is a continuous function. When $q$ or
$l$ is tuned larger, the discontinuity emerges, which could not observed in the
neutral AdS$_3$ backgrounds. We show that, the emergence of this discontinuity
is a universal behavior in the charged massive BTZ theory. With the increase of
graviton mass, on the other hand, no emergence of the discontinuity behavior
for any small $q$ and $l$ could be observed. We also show that the evolution of
the HEE and HC both become stable at later times, and $m_g$ speeds up reaching
to the stability during the evolution of the system. Moreover, we show that
$m_g$ decreases the final stable value of HEE but raises the stable value of
HC. Additionally, contrary to the usual picture in the literature that the
evolution of HC has only one peak, for big enough widths, we show that graviton
mass could introduce two peaks during the evolution. However, for large enough
charges the one peak behavior will be recovered again. We also examine the
evolutions of HEE and HC growths at the early stage, which an almost linear
behavior has been detected.
