We have studied the kinetics of thermoelastic martensitic transformations (MT), both experimentally and using spin models. Measurements of acoustic emission and thermal pulses in Cu-Zn-A1 single crystals, during thermally and stress induced MT, show that the transformation takes place as a sequence of avalanches, whose sizes and durations distribute according to power laws in more than one decade. Most features observed are qualitatively reproduced by spin models that incorporate domain interactions and frozen disorder. A systematic simulation of these models at T = 0 (no fluctuations allowed) has shown that there is a critical concentration of disorder for which avalanche sizes and durations distribute following power-laws, which appear to be independent of details of the models other than their dimensionality. We discuss the connection between the experimental results and the simulations.