Abstract This article generalizes the small noise cutoff phenomenon obtained recently by Barrera, Högele and Pardo (JSP2021) to the mild solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and Lévy noises in the Wasserstein distance. The methods rely on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the multiplicative stochastic solution flows in terms of stochastic exponentials.