Academic migration is the change of host institution by a researcher, typically aimed at achieving a stronger research profile. Scientific features such as the number of collaborations, the productivity and its research impact tend to be directly affected by such movements. In this paper, we analyse the dynamics of the collaboration network of researchers as they move from an institution to the next one. We specifically highlight cases where they increase and when they shrink, and quantify the dependency between the collaboration networks before and after such a movement. Finally, we drill down the analysis by dividing movements depending on the career stage of the researchers. The analysis shows a remarkable dynamism of collaboration networks across migrations. Interestingly, not always movements result in larger collaboration networks, while the overall similarity between networks across movements is quite limited on average. Qualitatively, the same effects can be found at all career stages, while, clearly, the magnitude of them might vary. These results are based on a dataset extracted from Scopus, containing detailed scientific information for the publications of 84,141 researchers.