We study diffusion on a multilayer network where the contact dynamics between the nodes is governed by a random process and where the waiting time distribution differs for edges from different layers. We study the impact on a random walk of the competition that naturally emerges between the edges of the different layers. In opposition to previous studies which have imposed a priori inter-layer competition, the competition is here induced by the heterogeneity of the activity on the different layers. We first study the precedence relation between different edges and by extension between different layers, and show that it determines biased paths for the walker. We also discuss the emergence of cyclic, rock-paper-scissors random walks, when the precedence between layers is non-transitive. Finally, we numerically show the slowing-down effect due to the competition on a heterogeneous multilayer as the walker is likely to be trapped for a longer time either on a single layer, or on an oriented cycle . Keywords: random walks; multilayer networks; dynamical systems on networks; models of networks; simulations of networks; competition between layers.