Recent studies of excitons in wide quantum wells have shown that the magnetic properties are strongly affected as the excitons acquire kinetic energy. In the center of mass approximation, these motion-induced changes are ascribed to mixing between the 1S exciton ground state and the higher lying nP states. The origin of the mixing is due to the dispersion curves for the valence band not being of simple parabolic form. Detailed previous studies of excitons in CdTe have resulted in excellent agreement between experiment and the predictions of this model. One consequence of the mixing is that the magnetic moment of the exciton is not simply the sum of the magnetic moments of the electron and hole, but contains motion-induced contributions, which can easily dominate the contributions from the individual charge carriers. To confirm the validity of the model, we have carried out detailed investigations of the magnetic properties of center of mass excitons in a second semiconductor, ZnSe, for which the magneto-optical properties of the individual charge carriers are completely different from those of CdTe. Excellent agreement is obtained between theory and experiment with a choice of the Luttinger parameter gamma(3)=0.98, in close agreement with the value determined independently by two-photon magnetoabsorption experiments. The success of the model when applied to both materials provides strong evidence that motion-induced changes in magnetism are a universal feature in zinc-blende semiconductors.