The new generation of storage-ring based synchrotron-radiation light sources imposes challenges on off-axis injection schemes owing to small dynamic apertures. In many upgrade projects of existing light sources, including the Swiss Light Source (SLS) upgrade, it is planned to reuse their injector booster synchrotrons with little to no modifications, although their emittances may not be suitable for the upgraded storage ring. A reduction of injection beam emittance relaxes the requirements for injection devices and machine aperture. Since the beam emittance in an electron synchrotron is much smaller in the vertical plane than in the horizontal, where injections are normally implemented, a significant improvement can be made by reducing the horizontal emittance by exchanging the transverse emittances through coupling resonance crossing. We find a meaningful scaling parameter for the quality of the emittance exchange, showing the relationship between coupling strength and crossing speed. Radiative effects are added to investigate the consequences for electron synchrotrons. Finally, we present the first experimental demonstration of transverse emittance exchange in the SLS booster synchrotron.