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

IllustrisTNG Snapshots for 10 Gyr of Dynamical Evolution of Brightest Cluster Galaxies and Their Host Clusters