Magnetic multilayers are promising tunable systems for hosting magnetic skyrmions at/above room temperature. Revealing their intriguing switching mechanisms and associated inherent electrical responses are prerequisites for developing skyrmionic devices. In this Letter, we theoretically demonstrate the annihilation of single skyrmions occurring through a multilayer structure, which is mediated by the hopping dynamics of topological hedgehog singularities known as Bloch points. The emerging intralayer dynamics of Bloch points are dominated by the Dzyaloshinskii-Moriya interaction, and their propagation can give rise to solenoidal emergent electric fields in the vicinity. Moreover, as the topology of spin textures can dominate their emergent magnetic properties, we show that the Bloch-point hopping through the multilayer will modulate the associated topological Hall response, with the magnitude proportional to the effective topological charge. We also investigate the thermodynamic stability of these states regarding the layer-dependent magnetic properties. This study casts light on the emergent electromagnetic signatures of skyrmion-based spintronics, rooted in magnetic-multilayer systems.