[abridged] Characterizing the number counts of faint, dusty star-forming galaxies is currently a challenge even for deep, high-resolution observations in the FIR-to-mm regime. They are predicted to account for approximately half of the total extragalactic background light at those wavelengths. Searching for dusty star-forming galaxies behind massive galaxy clusters benefits from strong lensing, enhancing their measured emission while increasing spatial resolution. Derived number counts depend, however, on mass reconstruction models that properly constrain these clusters. We estimate the 1.1 mm number counts along the line of sight of three galaxy clusters, i.e. Abell 2744, MACSJ0416.1-2403 and MACSJ1149.5+2223, which are part of the ALMA Frontier Fields Survey. We perform detailed simulations to correct these counts for lensing effects. We use several publicly available lensing models for the galaxy clusters to derive the intrinsic flux densities of our sources. We perform Monte Carlo simulations of the number counts for a detailed treatment of the uncertainties in the magnifications and adopted source redshifts. We find an overall agreement among the number counts derived for the different lens models, despite their systematic variations regarding source magnifications and effective areas. Our number counts span ~2.5 dex in demagnified flux density, from several mJy down to tens of uJy. Our number counts are consistent with recent estimates from deep ALMA observations at a 3$sigma$ level. Below $approx$ 0.1 mJy, however, our cumulative counts are lower by $approx$ 1 dex, suggesting a flattening in the number counts. In our deepest ALMA mosaic, we estimate number counts for intrinsic flux densities $approx$ 4 times fainter than the rms level. This highlights the potential of probing the sub-10 uJy population in larger samples of galaxy cluster fields with deeper ALMA observations.
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