With the advent of high-energy x-ray sources it is now possible to follow microstructural evolution in three dimensions and as a function of time. The ability to observe and quantify the evolution of a microstructure provides fundamentally new insights into this complex process. This is especially true of those microstructures that have complicated interfacial morphologies, such as the two-phase mixtures that are produced following dendritic solidification. The evolution of dendritic solid-liquid mixtures is determined using in situ three-dimensional (4D) x-ray tomography. Through these experiments it is possible to identify the mechanisms responsible for the evolution of the structure and, since the locations of the interfaces are known, to explore quantitatively the relationships between the dynamics of interfacial evolution and interfacial morphology. In this two-phase mixture, as well as others such as polymer blends, a region of one phase can fission into two by interfacial energy driven diffusion. The evolution of solid-liquid interfaces near these topological singularities has been investigated by examining the pinching of a rod of one phase embedded in another. We show theoretically and verify through 4D experiments that sufficiently close to the pinching event the interfacial morphology becomes universal: the interface shape is independent of the initial morphology of the rod-like phase and material system, and a power law describes its temporal evolution. This universality allows the dynamics of capillary driven microstructural break-up processes to be predicted in a vast array of materials, from steels to non-crystalline materials such as polymer blends.