  [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.
