This paper describes full solutions of the energy and Navier–Stokes equations in the approximate form of Boussinesq. The advective fluid layer flowing within parallel horizontal infinite walls subject to hydro-thermal slip conditions is of the prime interest. The control of the momentum/thermal motion is undertaken by a vertically applied magnetic field towards the parallel walls. The response of the layer to the momentum slip and thermal jump conditions under the applied magnetic field is investigated through solving exactly the idealized system of equations. From the obtained closed-form formulae, behaviour of the velocity and temperature fields as well as the rigid/free and thermally conducting/insulating wall cases is easy to gain. Results clearly imply that hydro-thermal slip enhances both velocity and temperature fields, unlike the suppression effects of magnetic field. Full solutions as presented here can serve as good basic flow for further research including the linear/nonlinear stability issues in regard to the plane or spiral perturbations.