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MmWaves have been envisioned as a promising direction to provide Gbps wireless access. However, they are susceptible to high path losses and blockages, which directional antennas can only partially mitigate. That makes mmWave networks coverage-limited, thus requiring dense deployments. Integrated access and backhaul (IAB) architectures have emerged as a cost-effective solution for network densification. Resource allocation in mmWave IAB networks must face big challenges to cope with heavy temporal dynamics, such as intermittent links caused by user mobility and blockages from moving obstacles. This makes it extremely difficult to find optimal and adaptive solutions. In this article, exploiting the distributed structure of the problem, we propose a Multi-Agent Reinforcement Learning (MARL) framework to optimize user throughput via flow routing and link scheduling in mmWave IAB networks characterized by user mobility and link outages generated by moving obstacles. The proposed approach implicitly captures the environment dynamics, coordinates the interference, and manages the buffer levels of IAB relay nodes. We design different MARL components, considering full-duplex and half-duplex IAB-nodes. In addition, we provide a communication and coordination scheme for RL agents in an online training framework, addressing the feasibility issues of practical systems. Numerical results show the effectiveness of the proposed approach.