Context. Cosmic dust is ubiquitous in astrophysical environments, where it significantly influences the chemistry and the spectra. Dust grains are likely to grow through the accretion of atoms and molecules from the gas-phase onto them. Despite their importance, only a few studies have computed the sticking coefficients for relevant temperatures and species, along with their direct impact on grain growth. Overall, the formation of dust and its growth are not well understood.Aims. This study is aimed at calculating the sticking coefficients, binding energies, and grain growth rates over a broad range of temperatures, for various gas species interacting with carbonaceous dust grains.Methods. We performed molecular dynamics simulations with a reactive force field algorithm to compute accurate sticking coefficients and obtain the binding energies. These results were used to build an astrophysical model of nucleation regions to study dust growth.Results. We present, for the first time, the sticking coefficients of H, H2 , C, O, and CO on amorphous carbon structures for temperatures ranging from 50 K to 2250 K. In addition, we estimated the binding energies of H, C, and O in carbonaceous dust to calculate the thermal desorption rates. Combining accretion and desorption allows us to determine an effective accretion rate and sublimation temperature for carbonaceous dust.Conclusions. We find that sticking coefficients can differ substantially from what is commonly used in astrophysical models. This offers us new insights into carbonaceous dust grain growth via accretion in dust-forming regions.