Climate change will impact cities' infrastructure and urban dwellers, who often show differentiated capacity to cope with climate-related hazards. The Shared Socioeconomic Pathways (SSPs) are part of an emerging research field which uses global socioeconomic and climate scenarios, developed by the climate change research community, to explore how different socioeconomic pathways will influence future society's ability to cope with climate change. While the SSPs have been extensively used at the global scale, their use at the local and urban scale has remained rare, as they first need to be contextualized and extended for the particular place of interest. In this study, we present and apply a method to develop multi-scale extended SSPs at the city and neighborhood scale. Using Boston, Massachusetts, as a case study, we combined scenario matching, experts' elicitation, and participatory processes to contextualize and make the global SSPs relevant at the urban scale. We subsequently employed the extended SSPs to explore future neighborhood-level vulnerability to extreme heat under multiple plausible socioeconomic trajectories, highlighting the usefulness of extended SSPs in informing future vulnerability assessments. The large differences in outcomes hint at the enormous potential of risk reduction that social and urban planning policies could trigger in the next decades.