Many biochemical systems are spatially heterogeneous and exhibit nonlinear behaviors, such as state switching in response to small changes in the local concentration of diffusible molecules. Systems as varied as blood clotting, intracellular calcium signaling, and tissue inflammation are all heavily influenced by the balance of rates of reaction and mass transport phenomena including flow and diffusion. Transport of signaling molecules is also affected by geometry and chemoselective confinement via matrix binding. In this review, we use a phenomenon referred to as patchy switching to illustrate the interplay of nonlinearities, transport phenomena, and spatial effects. Patchy switching describes a change in the state of a network when the local concentration of a diffusible molecule surpasses a critical threshold. Using patchy switching as an example, we describe conceptual tools from nonlinear dynamics and chemical engineering that make testable predictions and provide a unifying description of the myriad possible experimental observations. We describe experimental microfluidic and biochemical tools emerging to test conceptual predictions by controlling transport phenomena and spatial distribution of diffusible signals, and we highlight the unmet need for in vivo tools.