We present extensive numerical studies of the crossover from three-dimensional to two-dimensional systems in the nonequilibrium zero-temperature random-field Ising model with metastable dynamics. Bivariate finite-size scaling hypotheses are presented for systems with sizes L×L×l which explain the size-driven critical crossover from two dimensions (l=const, L→) to three dimensions (L→). A model of effective critical disorder Rceff(l,L) with a unique fitting parameter and no free parameters in the Rceff(l,L→) limit is proposed, together with expressions for the scaling of avalanche distributions bringing important implications for related experimental data analysis, especially in the case of thin three-dimensional systems. © 2018 American Physical Society.