We study the sedimentation of an initially inhomogeneous distribution of binary colloidal mixtures confined to a slit using a coarse-grained hybrid molecular dynamics and stochastic rotation dynamics simulation technique. This technique allows us to take into account the Brownian motion and hydrodynamic interactions between colloidal particles in suspensions. The sedimentation of such systems results in the formation of Rayleigh-Taylor-like hydrodynamic instabilities, and here we examine both the process of the formation and the evolution of the instability, as well as the structural organization of the colloids, depending on the properties of the binary mixture. We find that the structural properties of the swirls that form as a consequence of the instability depend greatly on the relative magnitudes of the Peclet numbers, and much less on the composition of the mixture. We also calculate the spatial colloid velocity correlation functions which allow us to follow the time evolution of the instability and the time dependence of the characteristic correlation length. Finally, we calculate the growth rates of the unstable modes both directly from our simulation data, and also using a theoretical approach, finding good agreement.