The physical models implemented in the recently developed dust dynamics code MIGRAINe are described. A major update of the treatment of secondary electron emission, stemming from models adapted to typical scrape-off layer temperatures, is reported. Sputtering and plasma species backscattering are introduced from fits of available experimental data and their relative importance to dust charging and heating is assessed in fusion-relevant scenarios. Moreover, the description of collisions between dust particles and plasma-facing components, based on the approximation of elastic-perfectly plastic adhesive spheres, has been upgraded to take into account the effects of particle size and temperature.