Recently we showed that the critical nonequilibrium relaxation in the Swendsen-Wang algorithm is widely described by the stretched-exponential relaxation of physical quantities in the Ising or Heisenberg models. Here we make a similar analysis in the Berezinsky-Kosterlitz-Thouless phase transition in the two-dimensional (2D) XY model and in the first-order phase transition in the 2D q=5 Potts model, and find that these phase transitions are described by the simple exponential relaxation and

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... r-law relaxation of physical quantities, respectively. We compare the relaxation behaviors of these phase transitions with those of the second-order phase transition in the 3D and 4D XY models and in the 2D q-state Potts models for 2 < q < 4, and show that the species of phase transitions can be clearly characterized by the present analysis. We also compare the size dependence of relaxation behaviors of the first-order phase transition in the 2D q=5 and 6 Potts models, and propose a quantitative criterion on "weakness" of the first-order phase transition.