Long-horizon motion forecasting for multiple autonomous robots is challenging due to non-linear agent interactions, compounding prediction errors, and continuous-time evolution of dynamics. Learned dynamics of such a system can be useful in various applications such as travel time prediction, prediction-guided planning and generative simulation. In this work, we aim to develop an efficient trajectory forecasting model conditioned on multi-agent goals. Motivated by the recent success of physics-guided deep learning for partially known dynamical systems, we develop a model based on neural Controlled Differential Equations (CDEs) for long-horizon motion forecasting. Unlike discrete-time methods such as RNNs and transformers, neural CDEs operate in continuous time, allowing us to combine physics-informed constraints and biases to jointly model multi-robot dynamics. Our approach, named PINCoDE (Physics-Informed Neural Controlled Differential Equations), learns differential equation parameters that can be used to predict the trajectories of a multi-agent system starting from an initial condition. PINCoDE is conditioned on future goals and enforces physics constraints for robot motion over extended periods of time. We adopt a strategy that scales our model from 10 robots to 100 robots without the need for additional model parameters, while producing predictions with an average ADE below 0.5 m for a 1-minute horizon. Furthermore, progressive training with curriculum learning for our PINCoDE model results in a 2.7X reduction of forecasted pose error over 4 minute horizons compared to analytical models.