We demonstrate the gradual shift from scale-free intermittent microplasticity to a scale-dependent behavior via the introduction of a variety of microstructural features within the Al-Cu binary alloy system. As long as the obstacles to dislocation motion remain shearable, the statistics of intermittent microplasticity has fat-tailed contributions. The introduction of incoherent precipitates leads to a complete transition from scale-free power-law scaling to an exponential and scale-dependent distribution. These results demonstrate how non-Gaussian interactions survive across different microstructures and further suggest that characteristic microstructural length scales and obstacle pinning-strengths are of secondary importance for the intermittency statistics, as long as dislocations can shear their local environment.