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We present Distributed Deep Deterministic Policy Gradient (3DPG), a multi-agent actor-critic (MAAC) algorithm for Markov games. Unlike previous MAAC algorithms, 3DPG is fully distributed during both training and deployment. 3DPG agents calculate local policy gradients based on the most recently available local data (states, actions) and local policies of other agents. During training, this information is exchanged using a potentially lossy and delaying communication network. The network therefore induces Age of Information (AoI) for data and policies. We prove the asymptotic convergence of 3DPG even in the presence of potentially unbounded Age of Information (AoI). This provides an important step towards practical online and distributed multi-agent learning since 3DPG does not assume information to be available deterministically. We analyze 3DPG in the presence of policy and data transfer under mild practical assumptions. Our analysis shows that 3DPG agents converge to a local Nash equilibrium of Markov games in terms of utility functions expressed as the expected value of the agents local approximate action-value functions (Q-functions). The expectations of the local Q-functions are with respect to limiting distributions over the global state-action space shaped by the agents' accumulated local experiences. Our results also shed light on the policies obtained by general MAAC algorithms. We show through a heuristic argument and numerical experiments that 3DPG improves convergence over previous MAAC algorithms that use old actions instead of old policies during training. Further, we show that 3DPG is robust to AoI; it learns competitive policies even with large AoI and low data availability.