The time-dependent covariant density functional theory in 3D lattice space has been developed and applied to investigate the microscopic dynamics of the linear-chain cluster states for carbon isotopes in the reactions 4He+8Be and 4He+10Be without any symmetry assumptions. By examining the density distribution and its time evolutions, the structure and dynamics of the linear-chain states are analyzed, and the quasiperiodic oscillations of the clusters are revealed. For 4He+8Be, the linear-chain states evolve to a triangular configuration and then to a more compact shape. In contrast, for 4He+10Be, the lifetime of the linear-chain states is much more prolonged due to the dynamical isospin effects by the valence neutrons which slow down the longitudinal oscillations of the clusters and persist the linear-chain states. The dependence of the linear chain survival time and dynamical isospin effects on impact parameters have been illustrated as well.