Dispersive Fourier transform (TS-DFT) technique opens a fascinating pathway to explore the ultrafast non-repetitive even, which has been employed to study the build-up process of mode-locked lasers. Here the whole evolution dynamics (from starting up to shutting down) of conventional soliton (CS), stretched pulse (SP) and dissipative soliton (DS) are investigated by using TS-DFT technique. The relaxation oscillation can be always observed before the formation of stable pulse operation, which is stemmed from the inherent advantage of the single-walled carbon nanotube. However, owing to the different pulse features, they exhibit the distinct evolution dynamics in the starting and shutting processes. Some critical phenomena are observed, including transient complex spectrum broadening and frequency-shift interaction of SP and picosecond pulses. These results could further deepen the understanding of the mode-locked fiber laser from the real-time point of view and is helpful for the laser design and applications.