We report on the characterisation of the dust activity and dynamical evolution of two faint active asteroids, P/2019 A4, and P/2021 A5, observed with the 10.4m GTC using both imaging and spectroscopy. Asteroid P/2019 A4 activity is found to be linked to an impulsive event occurring some $pm$10 days around perihelion, probably due to a collision or a rotational disruption. Its orbit is stable over 100 Myr timescales. Dust tail models reveal a short-term burst producing (2.0$pm$0.7)$times$10$^6$ 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 $sim$50 days after perihelion, lasting $sim$5 to $sim$60 days, and ejecting (8$pm$2)$times$10$^6$ 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 timescales. 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 ($sim$0.2 m/s) are consistent with escape speeds from those small-sized objects.
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