Growth of hard--rod monolayers via deposition is studied in a lattice model using rods with discrete orientations and in a continuum model with hard spherocylinders. The lattice model is treated with kinetic Monte Carlo simulations and dynamic density functional theory while the continuum model is studied by dynamic Monte Carlo simulations equivalent to diffusive dynamics. The evolution of nematic order (excess of upright particles, standing--up transition) is an entropic effect and is mainly governed by the equilibrium solution, {rendering a continuous transition} (paper I, J. Chem. Phys. 145, 074902 (2016)). Strong non--equilibrium effects (e.g. a noticeable dependence on the ratio of rates for translational and rotational moves) are found for attractive substrate potentials favoring lying rods. Results from the lattice and the continuum models agree qualitatively if the relevant characteristic times for diffusion, relaxation of nematic order and deposition are matched properly. Applicability of these monolayer results to multilayer growth is discussed for a continuum--model realization in three dimensions where spherocylinders are deposited continuously onto a substrate via diffusion.
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