This paper investigates the factors that influence the international mobility of research scientists using a new measure of mobility derived from changes in affiliations reported by publishing scientists in a major global index of scholarly publications over the period 1996-2011. Using a gravity-based empirical framework, our research shows that measures of geographic and socioeconomic and scientific distance correlate negatively with scientist mobility between two countries. Scientific collaboration appears to be a major factor associated with the mobility of scientists. The analysis shows that the mobility of scientists particularly relies on flows of tertiary-level students in the opposite direction, from destination to origin country. This provides strong evidence that brain circulation is a complex and multi-directional phenomenon. For a majority of country pairs (dyads) in our sample, the mobility of scientists is generally better described by commensurate knowledge flows in both directions, rather than one dominating the other. The analysis also shows that mobility can be positively influenced by convergence in economic conditions and resources dedicated to R&D, as well as reduced visa-related restrictions.