Diffractive optical neural networks (DONNs) have emerged as a promising optical hardware platform for ultra-fast and energy-efficient signal processing. However, previous experimental demonstrations of DONNs have only been performed using coherent light, which is not present in the natural world. Here, we study the role of spatial optical coherence in DONN operation. We propose a numerical approach to efficiently simulate DONNs under input illumination with arbitrary spatial coherence and discuss the corresponding computational complexity using coherent, partially coherent, and incoherent light. We also investigate the expressive power of DONNs and examine how coherence affects their performance. We show that under fully incoherent illumination, the DONN performance cannot surpass that of a linear model. As a demonstration, we train and evaluate simulated DONNs on the MNIST dataset using light with varying spatial coherence.