The atomic processes in molecular beam epitaxy of InAs on the InAs(137) surface are investigated by means of first-principles total-energy calculations. We consider layer-by-layer growth on InAs(137) facets as a typical process during the evolution of shallow InAs islands in the Stranski-Krastanov growth mode of InAs on GaAs that is exploited for the self-assembly of heteroepitaxial quantum dots. From the calculated energetics we conclude that a growth scenario where an As_2 molecule adsorbs on a single In adatom, followed by capture of another In adatom, is most likely. Moreover, our calculations of the potential-energy surface for In adatoms on the InAs(137) surface show that In adatoms are highly mobile. Surface diffusion on InAs(137) is found to be almost isotropic with energy barriers <0.3eV for adatom hopping. Aiming at an understanding of the growth processes at the strained side facets of quantum dots, we extend our calculations to isotropically strained InAs(137) facets. It is found that the compressive strain present on side facets of shallow InAs islands on GaAs leads to a considerable lowering of the binding energy of In adatoms. The height of diffusion barriers is found to be less affected by the strain. Most importantly, the intermediate species consisting of an In adatom plus an adsorbed As_2 molecule is destabilized by compressive strain in excess of -5%. This finding leads us to the conclusion that layer growth on InAs(137) facets ceases in highly strained regions of InAs islands on GaAs, in line with the observed shape evolution of such islands.