We consider the slow motion of viscous fluid completely filling a rectangular container. The motion is generated by the combined action of differential wall temperatures and the linear motion of the lid. If the relevant Reynolds and Péclet numbers and the lid speed are all small enough, the velocity field will be governed by an inhomogeneous biharmonic equation. In this approximation the temperature field, unaffected by the fluid motion, drives, at least in part, the fluid velocity field. Of interest here are the relative effects of buoyancy and lid motion. It is shown that the field, suitably scaled, depends on the dimensionless depth and lid speed alone. The mixed convection problem is solved for two pairs of wall heating protocols by sequentially solving, by an eigenfunction expansion method, up to four biharmonic problems. We present streamline patterns and quantitative data on the relative effects of lid motion on the buoyancy-driven fields in these containers.