The article proposes a new method for studying academic mobility, which, based on the author's publications, builds a history of his affiliation changing places, cities and countries of residence. This method was applied for studying academic mobility for publications of Russian researchers indexed in the Scopus database in 2000–2019. The results demonstrate that the depressive process of brain drain in Russia in 2015 was transformed into a process of brain circulation, when the number of immigrants in science became comparable to that of emigrants. In 2007–2011, most of the scientists left for the USA, Germany and other European countries. In 2015–2016, the number of scientists, who came to Russia from the CIS countries, approximately doubled, which compensated for the outflow in other areas. The intra-Russian migration of researchers is 76% related to Moscow, where more researchers come than leave the regions. In addition to the capital, St. Petersburg stands out, where there is a steady influx of researchers from other regions, which compensates for those leaving for the capital, and the Novosibirsk Region, which is the center of migration processes in the Ural and Siberian federal districts and the largest donor for other regions. At the same time, in all three regions, the outflow of researchers abroad in 2012–2016 significantly exceeded the inflow from other regions. In the rest of the Russian Federation, academic mobility in the indicated years was markedly less pronounced.