We report on a study of the time-resolved luminescence of type-II GaAs/AlAs superlattices near the type-I to type-II crossover. In spite of the slight type-II band alignment, the luminescence is dominated by the type-I transition. This is due to the inhomogeneous broadening of the type-I transition and the weak type-II oscillator strength, leading to a dominant luminescence of deeply localized type-I states. The relaxation within these localized states is found to be in agreement with a hopping model. At higher temperatures and excitation densities, the AlAs X minima act as an electron reservoir, leading to a strong increase of the luminescence decay time.