A stressed thin film on a soft substrate can develop complex wrinkle patterns. The onset of wrinkling and initial growth is well described by a linear perturbation analysis, and the equilibrium wrinkles can be analyzed using an energy approach. In between, the wrinkle pattern undergoes a coarsening process with a peculiar dynamics. By using a proper scaling and two-dimensional numerical simulations, this paper develops a quantitative understanding of the wrinkling dynamics from initial growth through coarsening till equilibrium. It is found that, during the initial growth, a stress-dependent wavelength is selected and the wrinkle amplitude grows exponentially over time. During coarsening, both the wrinkle wavelength and amplitude increases, following a power-law scaling under uniaxial compression. More complicated dynamics is predicted under equibiaxial stresses, which starts with a faster coarsening rate before asymptotically approaching the same scaling under uniaxial stresses. At equilibrium, a parallel stripe pattern is obtained under uniaxial stresses and a chaotic labyrinth pattern under equibiaxial stresses. Under stresses of the same magnitude, the two patterns have the same average wavelength, but different amplitudes. It is noted that the dynamics of wrinkling, while analogous to other phase ordering phenomena, is distinct and rich under the effects of stress and substrate elasticity.