This article aims to complexify the linguistic dimension of international schooling in light of the increasing diversification of the field but also as a result of the 'banalisation' of English and the growing 'added' value of multilingual competence in the knowledge economy. Drawing on data from focus groups with mobile families and institutional documents from an international school in Switzerland, we claim that the value of English-medium education to facilitate worldwide mobility is simultaneously conceived by parents as an obstacle for their children's acquisition of certain linguistic capitals in the locality. This engenders constant family (re-)evaluations of school choice and the development of strategies for children to acquire locally-available linguistic competences. The Swiss context activates the multilingual imagination of global middle-class families who demand that the school help their children maximise their chances of local linguistic capitalisation. In our case study, the current educational shift responds to parental desires for elite multilingualism with French and materialises in an optional dual-language programme to attract an increasing number of Swiss and established transnational families in a competitive eduscape.