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Harvesting data from distributed Internet of Things (IoT) devices with multiple autonomous unmanned aerial vehicles (UAVs) is a challenging problem requiring flexible path planning methods. We propose a multi-agent reinforcement learning (MARL) approach that, in contrast to previous work, can adapt to profound changes in the scenario parameters defining the data harvesting mission, such as the number of deployed UAVs, number, position and data amount of IoT devices, or the maximum flying time, without the need to perform expensive recomputations or relearn control policies. We formulate the path planning problem for a cooperative, non-communicating, and homogeneous team of UAVs tasked with maximizing collected data from distributed IoT sensor nodes subject to flying time and collision avoidance constraints. The path planning problem is translated into a decentralized partially observable Markov decision process (Dec-POMDP), which we solve through a deep reinforcement learning (DRL) approach, approximating the optimal UAV control policy without prior knowledge of the challenging wireless channel characteristics in dense urban environments. By exploiting a combination of centered global and local map representations of the environment that are fed into convolutional layers of the agents, we show that our proposed network architecture enables the agents to cooperate effectively by carefully dividing the data collection task among themselves, adapt to large complex environments and state spaces, and make movement decisions that balance data collection goals, flight-time efficiency, and navigation constraints. Finally, learning a control policy that generalizes over the scenario parameter space enables us to analyze the influence of individual parameters on collection performance and provide some intuition about system-level benefits.