This paper proposes a novel control framework for cooperative transportation of cable-suspended loads by multiple unmanned aerial vehicles (UAVs) operating in constrained environments. Leveraging virtual tube theory and principles from dissipative systems theory, the framework facilitates efficient multi-UAV collaboration for navigating obstacle-rich areas. The proposed framework offers several key advantages. (1) It achieves tension distribution and coordinated transportation within the UAV-cable-load system with low computational overhead, dynamically adapting UAV configurations based on obstacle layouts to facilitate efficient navigation. (2) By integrating dissipative systems theory, the framework ensures high stability and robustness, essential for complex multi-UAV operations. The effectiveness of the proposed approach is validated through extensive simulations, demonstrating its scalability for large-scale multi-UAV systems. Furthermore, the method is experimentally validated in outdoor scenarios, showcasing its practical feasibility and robustness under real-world conditions.