The transportation of large payloads can be made possible with Multi-Robot Systems (MRS) implementing cooperative strategies. In this work, we focus on the coordinated MRS trajectory planning task exploiting a Model Predictive Control (MPC) framework addressing both the acting robots and the transported load. In this context, the main challenge is the possible occurrence of a temporary mismatch among agents’ actions with consequent formation errors that can cause severe damage to the carried load. To mitigate this risk, the coordination scheme may leverage a leader–follower approach, in which a hierarchical strategy is in place to trade-off between the task accomplishment and the dynamics and environment constraints. Nonetheless, particularly in narrow spaces or cluttered environments, the leader’s optimal choice may lead to trajectories that are infeasible for the follower and the load. To this aim, we propose a feasibility-aware leader–follower strategy, where the leader computes a reference trajectory, and the follower accounts for its own and the load constraints; moreover, the follower is able to communicate the trajectory infeasibility to the leader, which reacts by temporarily switching to a conservative policy. The consistent MRS co-design is allowed by the MPC formulation, for both the leader and the follower: here, the prediction capability of MPC is key to guarantee a correct and efficient execution of the leader–follower coordinated action. The approach is formally stated and discussed, and a numerical campaign is conducted to validate and assess the proposed scheme, with respect to different scenarios with growing complexity.