Significance Space–time-dependent electromagnetic media, whose constitutive parameters are directionally biased in space and time, are a topic of current enormous interest across physics. Here we address a problem of fundamental relevance: the similarities and differences between such modulated materials and the classic problem of moving media. We establish that, surprisingly, the Fresnel drag effect of light in moving media, dictated by special relativity, is also present in modulated media, fully resolving this paradox with an analytical approach. We discuss how nonreciprocity can emerge at vanishing frequency. While we focus on electromagnetism, our findings can be extended to other wave systems.