Most of the biological polymers that make up our cells and tissues are hierarchically structured. For biopolymers ranging from collagen, to actin, to fibrin, this hierarchy provides vitally important versatility, allowing a multitude of structurally and functionally distinct structures to be constructed from a limited set of biomolecular constituents. This structural hierarchy must be encoded in the self-assembly process, from the earliest stages onward, in order to produce the appropriate substructures in the correct sequence. In this Letter, we explore the kinetics of such multi-stage self-assembly processes in a model system which is formulated as a set of discrete master equations capturing the underlying hierarchical molecular-scale process, but which may be homogenized to yield a practical, continuum description in terms of PDEs to compare to bulk experiments such as light scattering or turbidity measurements. We present the general framework, and apply it to recent turbidimetry data on the self-assembly of collagen fibrils. Furthermore, our analysis suggests a connection between diffusion-limited aggregation kinetics and fibril growth, supported by slow, power-law growth at very long timescales observed in both systems.