We study the driving-rate and temperature dependence of the power-law exponents that characterize the avalanche distribution in first-order phase transitions. Measurements of acoustic emission in structural transitions in Cu-Zn-Al and Cu-Al-Ni are presented. We show how the observed behavior emerges within a general framework of competing time scales of avalanche relaxation, driving rate, and thermal fluctuations. We confirm our findings by numerical simulations of a prototype model.