The first-order diffusionless structural transition in Cu67.64Zn16.71Al15.65 is characterized by jerky propagation of phase fronts related to the appearance of avalanches. In this paper, we describe a full analysis of this avalanche behavior using calorimetric heat-flux measurements and acoustic emission measurements. Two different propagation modes, namely, smooth front propagation and jerky avalanches, were observed in extremely slow measurements with heating and cooling rates as low as a few 10−3 K/h. Avalanches show criticality where each avalanche leads to a spike in the heat flux. Their statistical analysis leads to a power law [P(E)∼E−ε, where P(E)dE is the probability to observe an avalanche with energy E in an interval between E and E+dE] with an energy exponent of ε=2.15±0.15 in excellent agreement with the results of acoustic emission measurements. Avalanches appear to be more common for heating rates faster than 5×10−3 K/h whereas smooth front propagation occurs in all calorimetric measurements and (almost) exclusively for slower heating rates. Repeated cooling runs were taken after a waiting time of 1 month (and an intermediate heating run). Correlations between the avalanche sequences of the two cooling runs were found for the strongest avalanche peaks but not for the full sequence of avalanches. The memory effect is hence limited to strong avalanches.