We report the first parameter-free calculations of surface diffusion of a cation adatom on a reconstructed As-stabilized GaAs(001)-(2×4) surface. It is found from the calculated migration potentials of cation adatoms that the long-bridge sites are the most favorable adsorption sites on the GaAs(001)-(2×4) surfaces at low adatom coverages. The calculated results for surface diffusion constants of Ga adatoms show that the Ga-adatom diffusion is anisotropic on the reconstructed GaAs(001) surfaces and that the direction of fast diffusion is parallel to the As-missing dimer rows. The Al-adatom diffusion exhibits anisotropy similar to that of the Ga-adatom diffusion, while Al adatoms diffuse several times more slowly than Ga adatoms in the same directions in spite of the lighter mass of Al. Incorporating the calculated results for diffusion constants and migration potentials, the dynamical behavior of cation adatoms on the GaAs(00l) surface are demonstrated by stochastic Monte Carlo simulations at finite temperatures.