Electrophoresis of a dielectric fluid droplet with arbitrary surface potentials is investigated theoretically in this study. A pseudo-spectral method based on Chebyshev polynomials is adopted to solve the governing electrokinetic equations. Droplet mobilities are expressed as functions of electrolyte strength for fluid droplets with various viscosities and surface potentials. Effects of various electrokinetic parameters upon the droplet motion are investigated extensively. It is found, among other things, that the viscosity-dependence of the droplet mobility is synchronized with the presence of a separate axisymmetric exterior vortex flow surrounding the droplet, in addition to the axisymmetric interior vortex flow induced by the spinning motion of the charged surface. With its presence, the more viscous a fluid droplet is, the faster it moves; otherwise, the viscosity-dependence is the other way around. Critical points are discovered for highly charged droplets, at which the droplet surface becomes immobile and the interior fluid becomes motionless, just like a rigid solid particle. The orientation of the interior vortex flow changes each time passing these critical points, accompanied by the onset or shutdown of the exterior vortex flow. The results are useful in various practical applications such as drug delivery.