Results of a x-ray photon correlation spectroscopy experiment on the very weakly first order martensitic transformation of a Au50.5Cd49.5 single crystal are presented. Slow non-equilibrium-dynamics are observed in a narrow temperature interval in the direct vicinity of the otherwise athermal phase transformation. These dynamics are associated with the martensite-aging effect. The dynamical aging is accompanied by an avalanchelike behavior which is identified with an incubation-time phenomenon.