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 is built on fully exploiting the unique dynamic couplings within the multilift system. It parallelizes gradient computation across quadrotors and focuses on actual system state sensitivities relative to key MPC parameters. Extensive simulations demonstrate 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. It also excels in learning an adaptive reference for reconfiguring the system when traversing multiple narrow slots.