Following the Russian invasion of Ukraine, many Russian citizens have left Russia due to increasing repressions by the government, the fear of mobilization, or to escape the economic downturn. As of yet, reliable statistical data on those who left are not available. Hence, much remains unknown about the characteristics and scope of this population. However, migrants in the digital age prepare for their journeys by searching online, and these digital traces provide clues about their intentions and the scale of the migration event. Here we aim to leverage this resource to study geographic, temporal, and demographic trends in international migration intentions of Russian residents. Our analysis combines search queries provided by Yandex Wordstat with city-level data on socio-demographic and geographic characteristics and examines the two exodus waves, the first one shortly after the war began and the second one shortly after mobilization efforts started. Extracting meaning from this data with a gravity modelling approach, we find that regional indicators for wage levels were less important for the second wave and the proximity of a country of interest increased fourfold in importance. These and other results support dominant narratives in the media about the Russian brain drain and the scale of the exodus. Beyond this specific study, we also argue that Yandex searches have the potential to inform migration research.