Recent studies showed both experimental and numerical evidence that the occurrence probability of freak waves could be significantly enhanced as results of non-equilibrium dynamics induced by strong depth variations. The sea-state is characterized by strong non-Gaussian behavior in a short spatial extent after the depth transition, covering a few wavelengths. In this work, we investigate the complete equilibration process of an out-of-equilibrium sea-state via high-fidelity numerical simulations. In the simulations, the region after the depth transition is set as long as around one hundred wavelengths, such that the spectral adaptation develops and terminates eventually. The results are analyzed with spectral, cross-spectral and statistical approaches. It is shown that there are two stages with different spatial scales in the equilibration process. In the short scale, the sea-state is characterized by significant changes in wave statistics, freak wave occurrence probability is intensified. In the long scale, the wave spectrum undergoes strong modulation, the spectral peak disintegrate into a relative broad band, and low-frequency waves are enhanced as well. We show evidence that the spectral changes in the long scale are due to interactions of free components. The wave nonlinearity is shown to be positively correlated to the magnitude of the dynamical responses, but irrelevant to the length of the spatial scales in the equilibration process. In the established shallow-water equilibrium, the freak wave occurrence probability becomes less than Gaussian expectation and the waves are asymmetric in the vertical direction and symmetric in the horizontal.