Inter-neuron communication delays are ubiquitous in physically realized neural networks such as biological neural circuits and neuromorphic hardware. These delays have significant and often disruptive consequences on network dynamics during training and inference. It is therefore essential that communication delays be accounted for, both in computational models of biological neural networks and in large-scale neuromorphic systems. Nonetheless, communication delays have yet to be comprehensively addressed in either domain. In this paper, we first show that delays prevent state-of-the-art continuous-time neural networks called Latent Equilibrium (LE) networks from learning even simple tasks despite significant overparameterization. We then propose to compensate for communication delays by predicting future signals based on currently available ones. This conceptually straightforward approach, which we call prospective messaging (PM), uses only neuron-local information, and is flexible in terms of memory and computation requirements. We demonstrate that incorporating PM into delayed LE networks prevents reaction lags, and facilitates successful learning on Fourier synthesis and autoregressive video prediction tasks.