Population models commonly use discrete structure classes to capture trait heterogeneity among individuals (e.g. age, size, phenotype, intracellular state). Upscaling these discrete models into continuum descriptions can improve analytical tractability and scalability of numerical solutions. Common upscaling approaches based solely on Taylor expansions may, however, introduce ambiguities in truncation order, uniform validity and boundary conditions. To address this, we introduce a discrete multiscale framework to systematically derive continuum approximations of structured population models. Using multiscale asymptotic methods applied to discrete systems, we identify regions of structure space for which a continuum representation is appropriate. The leading-order dynamics are governed by nonlinear advection in the bulk, with diffusive boundary-layer corrections near wavefronts and stagnation points. We also derive discrete descriptions for regions where a continuum approximation is fundamentally inappropriate. This multiscale framework can be applied to other heterogeneous systems with discrete structure to obtain appropriate upscaled dynamics with asymptotically consistent boundary conditions.