In classification tasks, it is crucial to meaningfully exploit the information contained in the data. While much of the work in addressing these tasks is focused on building complex algorithmic infrastructures to process inputs in a black-box fashion, little is known about how to exploit the various facets of the data before inputting this into an algorithm. Here, we focus on this latter perspective by proposing a physics-inspired dynamical system that adapts optimal transport principles to effectively leverage color distributions of images. Our dynamics regulates immiscible fluxes of colors traveling on a network built from images. Instead of aggregating colors together, it treats them as different commodities that interact with a shared capacity on the edges. The resulting optimal flows can then be fed into standard classifiers to distinguish images in different classes. We show how our method can outperform competing approaches on image classification tasks in datasets where color information matters.