We study international mobility in academia, with a focus on the migration of published researchers to and from Russia. Using an exhaustive set of over $2.4$ million Scopus publications, we analyze all researchers who have published with a Russian affiliation address in Scopus-indexed sources in 1996-2020. The migration of researchers is observed through the changes in their affiliation addresses, which altered their mode countries of affiliation across different years. While only $5.2\%$ of these researchers were internationally mobile, they accounted for a substantial proportion of citations. Our estimates of net migration rates indicate that while Russia was a donor country in the late 1990s and early 2000s, it has experienced a relatively balanced circulation of researchers in more recent years. These findings suggest that the current trends in scholarly migration in Russia could be better framed as brain circulation, rather than as brain drain. Overall, researchers emigrating from Russia outnumbered and outperformed researchers immigrating to Russia. Our analysis on the subject categories of publication venues shows that in the past 25 years, Russia has, overall, suffered a net loss in most disciplines, and most notably in the five disciplines of neuroscience, decision sciences, mathematics, biochemistry, and pharmacology. We demonstrate the robustness of our main findings under random exclusion of data and changes in numeric parameters. Our substantive results shed light on new aspects of international mobility in academia, and on the impact of this mobility on a national science system, which have direct implications for policy development. Methodologically, our novel approach to handling big data can be adopted as a framework of analysis for studying scholarly migration in other countries.