We present an experimental study of phase-transition avalanches during the stress-induced formation of martensite in a Cu-Zn-Al alloy, with particular attention to the effect of cycling. We have analyzed statistically the amplitudes, durations, and energies of the thermal events accompanying the transition, and found that these magnitudes distribute according to power laws in the first fifty cycles, within experimental error, with exponents α = 2.3 ± 0 . 2 , τ = 2.9 ± 0 . 7 , and ε = 1.8 ± 0 . 3 , respectively. However, the extent of power-law behavior is reduced by at least one decade after five hundred transition cycles, and the system is seen to become progressively subcritical. We present a method of measuring the distribution of energy barriers encountered along the transition, which in our case is found to spread over more than four orders of magnitude. Finally, we have observed that repeated cycling leads to a statistical reproducibility of transformation trajectories, and to the remarkable correlation between the cycle-to-cycle trajectory fluctuations and the cycle-averaged response function of the system observed recently in magnetic systems [J.S. Urbach, R.C. Madison, and J.T. Markert, Phys. Rev. Lett. 75, 4694 (1995)].