The Rayleigh-B\'enard convection problem deals with the motion of a compressible fluid in a tunnel heated from below and cooled from above. In this context, the so-called Boussinesq relation is used, claiming that the density deviation from a constant reference value is a linear function of the temperature. These density and temperature deviations then satisfy the so-called Oberbeck-Boussinesq equations. The rigorous derivation of this system from the full compressible Navier-Stokes-Fourier system was done by Feireisl and Novotn\'y for conservative boundary conditions on the fluid's velocity and temperature. In this talk, we investigate the derivation for Dirichlet boundary conditions, and show that differently to the case of conservative boundary conditions, the limiting system contains an unexpected non-local temperature term. This is joint work with Peter Bella (TU Dortmund) and Eduard Feireisl (CAS).