The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. In work with Martin Bauer, Facundo Mémoli and Mao Nishino, we consider a vast generalization of the notion of a metric measure space: for an arbitrary metric space Z, we define a Z-network to be a measure space endowed with a kernel valued in Z. We introduce a method for comparing Z-networks by defining a generalization of GW distance, which we refer to as Z-Gromov-Wasserstein (Z-GW) distance. In this talk, I will explain several general geometric and topological properties of Z-GW distances. I will then focus on applications to shape analysis, and specifically to "shape graphs", or graph structures whose edges are endowed with a notion of shape. Examples include filamentary structures such as arterial networks and road networks.