The relationship between structure and dynamics in a polystyrene-polyisoprene block copolymer melt in the vicinity of the order-disorder transition was studied by small-angle X-ray scattering (SAXS), X-ray photon correlation spectroscopy (XPCS), and rheology. Rheological measurements on the disordered state indicate the presence of a fast process arising from the relaxation of polyisoprene chains and a slow process resulting predominantly from the relaxation of concentration fluctuations. In contrast, XPCS measurements of the disordered phase are dominated by diffusion of micelles. Time-resolved SAXS. XPCS, and rheology experiments on samples quenched from disorder-to-order reveal the existence of two regimes. While the microscopic relaxation time, measured by XPCS, increases after all of the quenches. SAXS and rheological signatures of ordering are only seen when the quench depth exceeds a critical value of 10 degrees C. For quenches 5 degrees C below the order-to-disorder transition temperature, no changes in the SAXS profiles and rheological properties are observed on experimental time scales. It is evident that nucleation barriers preclude the formation of the ordered phase during shallow quenches. The time-resolved rheology measurements enable estimation of the nucleation barriers that are responsible for the observations in the shallow quench regime.