We examined the melt and solid-state structures of a series of diblock copolymers containing polyethylene as the minority block, with a rubbery hydrocarbon majority block. When the interblock segregation strength during crystallization is sufficiently high (approximately 3 times the segregation strength at the order-disorder transition), crystallization can be effectively confined within spherical domains formed by microphase separation in the melt; the process is homogeneously nucleated, and the resulting kinetics are first-order (Avrami n = 1). Below this critical interblock segregation strength, crystallization disrupts the spherical microdomains, resulting in sigmoidal kinetics (n > 1). Cylinder-forming materials are more complex: there exists a range of intermediate segregation strength where crystallization is templated but not wholly confined within the nanoscale domains prescribed by microphase separation; while the melt morphology is generally retained on cooling, local distortions and connections between cylinders occur due to crystallization. These intercylinder connections allow the material initially contained within several cylinders to be crystallized by a single nucleus, producing sigmoidal kinetics and a dramatic acceleration of the overall crystallization rate, despite the general preservation of the cylindrical structure.