  We explore the redshift evolution of the dynamical properties of massive
clusters and their brightest cluster galaxies (BCGs) at $z < 2$ based on the
IllustrisTNG-300 simulation. We select 270 massive clusters with $M_{200} <
10^{14}~{\rm M}_{\odot}$ at $z = 0$ and trace their progenitors based on merger
trees. From 67 redshift snapshots covering $z < 2$, we compute the 3D subhalo
velocity dispersion as a cluster velocity dispersion ($\sigma_{\rm cl}$). We
also calculate the 3D stellar velocity dispersion of the BCGs ($\sigma_{\rm
*,~BCG}$). Both $\sigma_{\rm cl}$ and $\sigma_{\rm *,~BCG}$ increase as
universe ages. The BCG velocity dispersion grows more slowly than the cluster
velocity dispersion. Furthermore, the redshift evolution of the BCG velocity
dispersion shows dramatic changes at some redshifts resulting from dynamical
interaction with neighboring galaxies (major mergers). We show that
$\sigma_{\rm *,~BCG}$ is comparable with $\sigma_{\rm cl}$ at $z > 1$, offering
an interesting observational test. The simulated redshift evolution of
$\sigma_{\rm cl}$ and $\sigma_{\rm *,~BCG}$ generally agrees with an observed
cluster sample for $z < 0.3$, but with large scatter. Future large
spectroscopic surveys reaching to high redshift will test the implications of
the simulations for the mass evolution of both clusters and their BCGs.
