We report on the characterization of the dust activity and dynamical evolution of two faint active asteroids, P/2019 A4, and P/2021 A5, observed with the 10.4 m GTC using both imaging and spectroscopy. Asteroid P/2019 A4 activity is found to be linked to an impulsive event occurring some ±10 d around perihelion, probably due to a collision or a rotational disruption. Its orbit is stable over 100 Myr time-scales. Dust tail models reveal a short-term burst producing (2.0 ± 0.7) × 106 kg of dust for maximum particle radius rmax = 1 cm. The spectrum of P/2019 A4 is featureless, and slightly redder than the Sun. P/2021 A5 was active ~50 d after perihelion, lasting ~5 to ~60 d, and ejecting (8 ± 2) × 106 kg of dust for rmax = 1 cm. The orbital simulations show that a few percent of dynamical clones of P/2021 A5 are unstable on 20-50 Myr time-scales. Thus, P/2021 A5 might be an implanted object from the JFC region or beyond. These facts point to water-ice sublimation as the activation mechanism. This object also displays a featureless spectrum, but slightly bluer than the Sun. Nuclei sizes are estimated in the few hundred meters range for both asteroids. Particle ejection speeds (≍0.2 m s-1) are consistent with escape speeds from those small-sized objects.