We describe microstructural aspects of phase transitions between the lamellar (L), perforated layer (PL), and gyroid (G) morphologies in diblock copolymer melts. Using small-angle scattering, dynamic mechanical spectroscopy, and transmission electron microscopy, we show that these transformations proceed through the nucleation and growth of the final phase, in contrast to recent calculations that assume evolution from a thermodynamically unstable initial state. Direct L --> G transitions are suppressed by the high surface tension associated with L-G grain boundaries; the formation of the metastable PL structure under such conditions reflects the ease with which the L --> PL transition can occur, compared to L --> G. Similar effects dominate the G --> L transition. Mismatches in spacings between epitaxially related lattice planes also influence relaxation kinetics; the P --> LG transition rate depends strongly on the relative spacings of the PL [10] and G [211] planes, and the considerable discrepancy between the G [211] and L [10] spacings at the L-G boundary may further retard that transformation. Similar factors have been shown to govern the evolution of amphiphilic systems, supporting geometrically inspired attempts to understand this phase behavior.