This article discusses three kinds of mobility among early stage researchers: geographical mobility, mobility between disciplines -- or interdisciplinarity -- and cross-sectoral mobility. It focuses on how PhD fellows engage with and negotiate experiences of mobility. These types of mobility have largely been presented as inherently beneficial in mainstream policy discourse, but this article presents a more nuanced picture of mobility, showing the challenges of mobility, as experienced and articulated by PhD fellows and some of their supervisors. The research is based on twenty-six interviews with PhD fellows and principal investigators involved in two types of flagship doctoral programmes: the ITN in Europe, and the IGERT in the United States. The main finding is that PhD fellows associated all three types of mobility with feelings of homelessness.