The charge and dynamics of dust particles in an afterglow plasma are studied using a 1D model in the diffusion approximation, taking into account the transition from ambipolar to free diffusion. It is analyzed how external conditions (dust particle size, neutral gas pressure and initial electron density) affect the dust motion. The dust particle dynamics has been examined in microgravity conditions and in presence of gravity. Without gravity, the location of dust particles in plasma volume may change essentially during the afterglow if the dust size and pressure are small (≤ 10 nm and ≤ 30 mTorr, respectively). At small pressures, in the very beginning of afterglow, small nanoparticles move to the plasma boundary because the ion drag force dominates over the electric force. At afterglow times when the electron temperature becomes time-independent, the ion drag force decreases faster with time than the electric force due to the ion density decrease, and dust particles may move to the slab center. In presence of gravity, the effect of gravity force on dust particles is important only at large afterglow times (t ≥ 10 ms), when the electric and ion drag forces are small. The dust dynamics depends essentially on the initial plasma density. If the density is large (~ 1012 cm-3), small nanoparticles (≤ 10 nm) may deposit on plasma walls in the beginning of plasma afterglow because of an enhancement of the ion drag force.