In this paper, we propose a kinetic description for the growth and evaporation of oxygen-rich, dirty dust particles, which consist of numerous small islands of different solid materials like Mg2SiO4, SiO2, Al2O3, Fe and TiO2. We assume that the total surface of such a grain collects condensible molecules from the gas phase and that these molecules are rapidly transported by diffusive hopping on the surface to the respective solid islands, where finally the constructive surface chemical reactions take place which increase the size of the grain. Applied to a typical dust forming region in a brown dwarf atmosphere, turbulent temperature fluctuations enable the creation of first seed particles (nucleation) at high supersaturation ratios. These seeds are then quickly covered by different solid materials in a simultaneous way, which results in dirty grains. Our treatment by moment equations allows for the calculation of the time-dependent material composition of the dust grains and the elemental composition of the gas phase. We argue that the depletion of condensible elements from the gas phase by dust formation may be incomplete and occurs in a patchy, non-uniform way which possibly makes metallicity measurements highly uncertain.