The design and control of winged aircraft and drones is an iterative process aimed at identifying a compromise of mission-specific costs and constraints. When agility is required, shape-shifting (morphing) drones represent an efficient solution. However, morphing drones require the addition of actuated joints that increase the topology and control coupling, making the design process more complex. We propose a co-design optimisation method that assists the engineers by proposing a morphing drone's conceptual design that includes topology, actuation, morphing strategy, and controller parameters. The method consists of applying multi-objective constraint-based optimisation to a multi-body winged drone with trajectory optimisation to solve the motion intelligence problem under diverse flight mission requirements, such as energy consumption and mission completion time. We show that co-designed morphing drones outperform fixed-winged drones in terms of energy efficiency and mission time, suggesting that the proposed co-design method could be a useful addition to the aircraft engineering toolbox.