We have investigated the pressure dependence of the room-temperature photoluminescence (PL) of selfassembled CdSe quantum dots (QDs) with quaternary ZnCdMgSe barriers. Four samples with different CdSe deposition times, i.e. different average dot size, were studied. Samples exhibit a very distinct emission energy covering the visible spectrum from the red to the blue. We found, however, a similar pressure coefficient of around 47 meV/GPa for the PL emission of all samples. This value is close to that of bulk CdSe, which gives evidence that dot and quaternary barrier material have a similar compressibility. In contrast, the PL linewidth displays a clear trend with respect to the emission energy, the PL line being broader for the smaller dots emitting in the blue. This is explained by taking into account the huge bandgap offset of ≈ 1 eV between dot and barrier and considering quantum confinement effects within a very simplistic infinitely deep square-well model.