We study products of precursors of spatially local operators, \(W_{x_{n}}(t_{n}) ... W_{x_1}(t_1)\), where \(W_x(t) = e^{-iHt} W_x e^{iHt}\). Using chaotic spin-chain numerics and gauge/gravity duality, we show that a single precursor fills a spatial region that grows linearly in \(t\). In a lattice system, products of such operators can be represented using tensor networks. In gauge/gravity duality, they are related to Einstein-Rosen bridges supported by localized shock waves. We find a geometrical correspondence between these two descriptions, generalizing earlier work in the spatially homogeneous case.