The configurations of steady two-dimensional flow accompanied by heat and mass transport in a shallow lid-driven cavity with a moving heated lid and a moving cooled lid were investigated numerically in a process engineering context of drying. A continuation method was applied to track the branches of the diagram of flow configurations in their dependence on various parameters and to determine their linear stability by an Arnoldi-based method. Analytical solutions for limiting situations of the geometrical and flow parameters, obtained by computer algebra, were compared with numerical results. In a parameter study for horizontal and vertical orientation of the cavity, the dependence of heat and mass transfer rates on the velocities of the walls and on the species concentration boundary conditions was investigated. Among the results found, for a vertical cavity, there are two turning points and flow configurations with minimal heat and mass transfer, and for a horizontal cavity heated from below a Hopf bifurcation indicating inception of oscillatory flow regimes.