Subgap states are responsible for the low-bias transport features of hybrid superconducting-semiconducting devices. Here we analyze the local and nonlocal differential conductance of Coulomb-blockaded multiterminal superconducting islands that host subgap states with different spatial structures. The emerging patterns of their transport spectroscopy are used to characterize the possible topological nature of these devices and offer the possibility of controlling their transport properties. We develop a next-to-leading order master equation to describe the multiterminal transport in superconductors with both strong Coulomb interactions and multiple subgap states, coupled with metallic leads. We show that the nonlocal differential conductance characterizes the spatial extension of the subgap states and signals the presence of degenerate bound states with a finite support on different parts of the device. Additionally, it displays sharp sign changes as a function of the induced charge of the superconductor, signaling energy crossings among its lowest excited states.