The extent to which international high-skilled mobility channels are forming is a question of great importance in an increasingly global knowledge-based economy. One factor facilitating the growth of high-skilled labour markets is the standardization of certifiable degrees meriting international recognition. Within this context, we analysed an extensive high-skilled mobility database comprising roughly 382 000 individuals from five broad profession groups (Medical, Education, Technical, Science & Engineering and Business & Legal) over the period 1997-2014, using the 13-country expansion of the European Union (EU) to provide insight into labour market integration. We compare the periods before and after the 2004 enlargement, showing the emergence of a new east-west migration channel between the 13 mostly eastern EU entrants (E) and the rest of the western European countries (W). Indeed, we observe a net directional loss of human capital from E → W, representing 29% of the total mobility after 2004. Nevertheless, the counter-migration from W → E is 7% of the total mobility over the same period, signalling the emergence of brain circulation within the EU. Our analysis of the country-country mobility networks and the country-profession bipartite networks provides timely quantitative evidence for the convergent integration of the EU, and highlights the central role of the UK and Germany as high-skilled labour hubs. We conclude with two data-driven models to explore the structural dynamics of the mobility networks. First, we develop a reconfiguration model to explore the potential ramifications of Brexit and the degree to which redirection of high-skilled labourers away from the UK may impact the integration of the rest of the European mobility network. Second, we use a panel regression model to explain empirical high-skilled mobility rates in terms of various economic 'push-pull' factors, the results of which show that government expenditure on education, per capita wealth, geographical proximity and labour force size are significant attractive features of destination countries.