While diffusion models can successfully generate data and make predictions, they are predominantly designed for static images. We propose an approach for training diffusion models for probabilistic dynamics forecasting that leverages the temporal dynamics encoded in the data, directly coupling it with the diffusion steps in the network. We train a stochastic, time-conditioned interpolator and a forecaster network that mimic the forward and reverse processes of conventional diffusion models, respectively. This design choice naturally encodes multi-step and long-range forecasting capabilities, allowing for highly flexible, continuous-time sampling trajectories and the ability to trade-off performance with accelerated sampling at inference time. In addition, the dynamics-informed diffusion process imposes a strong inductive bias, allowing for improved computational efficiency compared to traditional Gaussian noise-based diffusion models. Our approach performs competitively on probabilistic evaluations for forecasting complex dynamics in sea surface temperatures, Navier-Stokes flows, and spring mesh systems.