We investigate the non-equilibrium evolution of ideal Brownian particles confined between two walls, following simultaneous quenches of the temperature and a constant external force. We compute (analytically and in numeric simulations) the post-quench dynamics of the density and the pressure exerted by the particles on the two walls perpendicular to the drift force. For identical walls, symmetry breaking associated with the drift gives rise to unequal particle densities and pressures on the two walls. While the pressure on one wall increases monotonically after the quench, on the other wall, depletion causes a non-monotonic dynamics with an overshooting at finite times, before the long-term steady-state value is reached. For walls immersed in a Brownian gas, the effective interaction force changes sign from repulsive at short times to attractive at late times. These findings have potential applications in various soft matter systems or fluids with charged Brownian particles, as well as carrier dynamics in semiconducting structures.