Modern AI models are increasingly being used as theoretical tools to study human cognition. One dominant approach is to evaluate whether human-derived measures are predicted by a model's output: that is, the end-product of a forward pass. However, recent advances in mechanistic interpretability have begun to reveal the internal processes that give rise to model outputs, raising the question of whether models might use human-like processing strategies. Here, we investigate the relationship between real-time processing in humans and layer-time dynamics of computation in Transformers, testing 20 open-source models in 6 domains. We first explore whether forward passes show mechanistic signatures of competitor interference, taking high-level inspiration from cognitive theories. We find that models indeed appear to initially favor a competing incorrect answer in the cases where we would expect decision conflict in humans. We then systematically test whether forward-pass dynamics predict signatures of processing in humans, above and beyond properties of the model's output probability distribution. We find that dynamic measures improve prediction of human processing measures relative to static final-layer measures. Moreover, across our experiments, larger models do not always show more human-like processing patterns. Our work suggests a new way of using AI models to study human cognition: not just as a black box mapping stimuli to responses, but potentially also as explicit processing models.