Single layers and superlattices of GaAsN/GaAs and InGaAsN/GaAs were grown using metalorganic molecular beam epitaxy with dimethylhydrazine, trimethylindium, triethylgallium, and conventional arsenic sources. Nitrogen incorporation into the solid was investigated as a function of the substrate temperature and fluxes. The nitrogen incorporation kinetics and growth mechanism have been modeled by assuming formation of an adduct arising from reactions between triethylgallium and dimethylhydrazine, while neglecting reactions between precursors of trimethylindium and dimethylhydrazine. The model accounts for the experimentally observed relationship between growth rates and nitrogen incorporation in GaAsN and InGaAsN. Our experiments show that the absolute arsenic flux and the As/N flux ratio play a critical role in the growth of single phase GaAsN.