In the last decade, there has been a significant rise in the number of global social movements, generating claims of a new networked era of global social movements, in which young people often play a central role. Young generations and social movements tend to identify with a diverse set of social and political issues on both a global and local scale, while often displaying a progressive and transversal outlook. Among these youth social movements, we find Fridays For Future, a youth climate organization noted for its unique scope and tactics as well as for illustrating the prominence of environmental issues in youth activism. Since 2018, the movement has progressively grown into a network composed of many local and national groups across the world. Beyond analysis of the global character of the movement and the profile and motivations of its activists, it seems opportune to look at how—at the organizational level—these local groups evolve and position themselves. Based on a social network and a content analysis of Fridays For Future-Barcelona’s Twitter account since its creation, this article explores the organization’s “glocal” and transversal dynamics, the relationship that might exist between these two, and the potential influence of the COVID-19 pandemic on these processes. The results show that, stemming from a global preoccupation for climate change, this Youth Climate Movement organization has been increasingly oriented towards local-level networking, anchoring its activism in local struggles and realities. They also show a positively-associated process—reinforced by the COVID-19 pandemic—of increasing transversality, reflecting aspirations for global justice and the idea of an interdependency between climate change and other social, political and economic issues.