We use the rigorous anyon-fermion mapping method to investigate the dynamic evolution of strongly interacting anyons confined in weak harmonic traps. Density profiles, momentum distributions and reduced integral density matrices are obtained for various statistical parameters. Anyon density profiles exhibit the same behavior regardless of the evolving statistical parameters. When the harmonic trap is abruptly turned off, the momentum distribution exhibits symmetric fermion-like behavior with long-time evolution. When the trap frequencies are quenched, the momentum distribution exhibits an asymmetric breathing mode during evolution. The reduced unitary density matrix shows dynamic symmetry breaking and reproduced behavior.