Structure-property relationships are usually discussed in the context of optimization of the equilibrium properties of heterogeneous materials. The logic and consequences of the effects of grain size and shape in metals and ceramics, porosity and particle size distributions in aggregates, and fiber/matrix arrangement details in fibrous composites are familiar examples. However, for generalized force-flux problems that may not be equilibrium systems, multiphysical property and performance transitions are sometimes observed as a function of microstructure variations, including transitions in electrochemical response (e.g., corrosion), electrical and dielectric response (e.g., conduction) and mechanical response (including brittle transitions and fracture initiation). The present paper constructs a "critical path" concept that generalizes the familiar concept of percolation to create a broad foundation for predicting transitions in physical properties and performance that often define design boundaries or performance transitions that signal the onset of unsteady behavior such as fracture or chemical breakdown. A broad range of applications to diverse technologies such as structural composites, ceramic membranes and heterogeneous containment materials are discussed, supported by a new foundation of conformal, multiphysics, multiscale modeling that unifies the philosophy of the subject.