Designing motion control and planning algorithms for multilift systems remains challenging due to the complexities of dynamics, collision avoidance, actuator limits, and scalability. Existing methods that use optimization and distributed techniques effectively address these constraints and scalability issues. However, they often require substantial manual tuning, leading to suboptimal performance. This paper proposes Auto-Multilift, a novel framework that automates the tuning of model predictive controllers (MPCs) for multilift systems. We model the MPC cost functions with deep neural networks (DNNs), enabling fast online adaptation to various scenarios. We develop a distributed policy gradient algorithm to train these DNNs efficiently in a closed-loop manner. Central to our algorithm is distributed sensitivity propagation, which parallelizes gradient computation across quadrotors, focusing on actual system state sensitivities relative to key MPC parameters. We also provide theoretical guarantees for the convergence of this algorithm. Extensive simulations show rapid convergence and favorable scalability to a large number of quadrotors. Our method outperforms a state-of-the-art open-loop MPC tuning approach by effectively learning adaptive MPCs from trajectory tracking errors and handling the unique dynamics couplings within the multilift system. Additionally, our framework can learn an adaptive reference for reconfigurating the system when traversing through multiple narrow slots.