Social-ecological systems (SES) are complex and adaptive, with processes spanning a vast range of scales, from behaviours of individuals to global economic and earth-systems processes. Addressing urgent sustainability challenges requires an understanding of cross-scale interactions in SES; how large-scale phenomena emerge from, and feedback onto, small-scale phenomena. In this work, we model renewable resource harvesting in a multiscale, hierarchical SES in which resource stock dynamics are coupled to social dynamics constrained by a social norm of non-excessive harvesting. Using an approach based on the renormalization group from statistical physics, we study how the effectiveness of resource management by social pressure changes with scale, by virtue of scaling trends in the resource and social dynamics. We then introduce cross-scale feedbacks and study effects resulting from parameter adjustments at different scales. We discuss implications of our findings for management and governance of social-ecological systems including agriculture, fisheries and climate.