Light emissions from cavitating liquids serve as a diagnostic tool for chemical activity, bubble collapse conditions, or excited species. Here we demonstrate the influence of mechanical stirring on sonoluminescence (SL) and sonochemiluminescence (SCL) emissions emerging in the presence of dissolved sodium salts and luminol in different sonicated liquids. In the systems investigated, driven in the 20-40 kHz range, stirring can change the spatial distribution of blue/white broadband SL emissions and of the orange sodium D-line emission, as well as their relative intensities. In many cases, an amplification of sodium emission is observed under stirring, but striking exceptions appear as well. SCL emission from luminol is mainly quenched by the mechanical agitation. The liquids under study comprise water, ethylene glycol, and phosphoric acid, all with dissolved argon or krypton, and partly xenon. From the stirring-induced changes in sonoluminescence and from high-speed video recordings, we try to draw conclusions on the actual effects of the induced rapid bulk liquid flow on the cavitating systems. Main effects seem to be enhanced gas diffusion and bubble separation or de-clustering, possibly by a separative impact of the stirring flow on populations of smaller and larger bubbles, respectively. As a consequence, large bubbles with extremely bright sonoluminescence flashes appear in phosphoric acid under stirring. Overall, a certain sensitivity of stirring effects on further system parameters has to be stated.