We present a model that describes the drying of solutions and colloidal dispersions from droplets confined between two circular plates. This confined geometry, proposed by Clément and Leng [Langmuir 20, 6538 (2004)], casts a perfect control of the evaporation conditions, and thus also of the concentration kinetics of the solutes in the droplet. Our model, based on simple transport equations for binary mixtures, describes the concentration process of the solute inside the droplet. Using dimensionless units, we identify the different numbers that govern the concentration field of the solute, and we detail how to extract kinetic and thermodynamic information on the binary mixture from such drying experiments. We finally discuss, using numerical resolution of the model and analytical arguments, several specific cases: dilute solutions, a colloidal hard sphere dispersion, and a binary molecular mixture.