- A surprisingly large number of systems in nature are thought to be governed by internal dynamics which causes avalanches of various sizes. In such systems energy, which is delivered from outside, is redistributed as a result of the occurrence of localized avalanches. Starting an avalanche requires that some threshold condition be satisfied. Random driving (energy input) brings the system into a strongly inhomogeneous state, so that the probability of triggering an avalanche in a large part of the system is small. In most physical systems energy redistribution may occur due to diffusive processes without avalanches. Diffusion also makes the system more uniform, making large avalanche triggering more probable. The observed behavior of a such system may crucially depend on the competition between diffusion and driving. In this paper, the effects of diffusive processes are investigated using a dissipative, isotropic one-dimensional model, in which avalanches can propagate in both directions. It is shown that the system behavior changes progressively as the diffusion rate increases. In the absence of diffusion, many small avalanches are triggered. Increasing the diffusion rate gradually suppresses these small avalanches and leads to the development of large, quasi-periodic bursts.