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Diffractive deep neural networks (D2NNs) define an all-optical computing framework comprised of spatially engineered passive surfaces that collectively process optical input information by modulating the amplitude and/or the phase of the propagating light. Diffractive optical networks complete their computational tasks at the speed of light propagation through a thin diffractive volume, without any external computing power while exploiting the massive parallelism of optics. Diffractive networks were demonstrated to achieve all-optical classification of objects and perform universal linear transformations. Here we demonstrate, for the first time, a "time-lapse" image classification scheme using a diffractive network, significantly advancing its classification accuracy and generalization performance on complex input objects by using the lateral movements of the input objects and/or the diffractive network, relative to each other. In a different context, such relative movements of the objects and/or the camera are routinely being used for image super-resolution applications; inspired by their success, we designed a time-lapse diffractive network to benefit from the complementary information content created by controlled or random lateral shifts. We numerically explored the design space and performance limits of time-lapse diffractive networks, revealing a blind testing accuracy of 62.03% on the optical classification of objects from the CIFAR-10 dataset. This constitutes the highest inference accuracy achieved so far using a single diffractive network on the CIFAR-10 dataset. Time-lapse diffractive networks will be broadly useful for the spatio-temporal analysis of input signals using all-optical processors.