The observation and the study of first-order phase transitions in cooperative spin-crossover (SCO) solids exhibiting hysteresis behaviours are of particular interest and currently constitute a burgeoning area in the field of bistable molecular materials. The understanding and the control of the transition mechanisms (nucleation and growth processes) and their dynamics within the hysteresis region appear to be a general and appealing problem from a fundamental point of view and for technological applications as well. This review reports on the recent progresses and most important findings made on the spatiotemporal dynamics of the spin transition in SCO solids, particularly through the universal nucleation and growth process. Both thermally induced and light-induced spin transitions are discussed. We open up this review to the central question of the evolution of the transition mechanisms and dynamics in SCO nano-objects, which constitute promising systems to reach ultra-fast switching, and the experimental issues inherent to such studies at the micro- and nanometric scale.