I INTRODUCTIONFunctional materials are based on the interplay of different ferroic properties like ferroelasticity, ferromagnetism, ferroelectricity, etc. In order to obtain a sufficiently large response to the external excitation, these materials are typically tuned so to cross a first-order phase transition in which one or several order parameters (strain, magnetization, polarization, etc…) exhibit a macroscopic discontinuity. It is thus important for applications to understand the dynamics of first-order phase transitions (FOPT) in such materials.FOPT in solids hardly occur at thermal equilibrium. Typically, the energy barriers involved in the transition are very large compared to thermal fluctuations so that the order parameters evolve following metastable trajectories. The transitions are then called athermal, and instead of being sharp like in ideal first-order phase transitions, they extend over a broad range of the driving parameter and show hysteresis. In many cases the hysteresis (or at least a part of it) cannot be suppressed by driving the system more slowly because it is not related to the fact that the system cannot respond instantaneously due to relaxational delay. This kind of hysteresis is usually called rateindependent hysteresis.The high energy barriers have two origins: on the one hand, real materials always exhibit some amount of quenched disorder that determines the nucleation sites and can thus strongly affect the metastable path. On the other hand, when one of the order parameters involved in the transition is strain (like in martensitic transformations), a