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The paper introduces a new, intrinsically discrete, path planning and collision-avoidance problem, with multiple robots and multiple goals. The issue arises in the operation of the novel Swing and Dock (SaD) locomotion for a material handling system. Its agents traverse a base grid by sequences of rotations (swings) around fixed pins. Each agent must visit an array of goal positions in minimal time while avoiding collisions. The corresponding off-line path-planning problem is NP-hard. We model the system by an extended temporal graph and introduce two integer linear programming (ILP) formulations for the minimization of the makespan, with decision variables on the nodes and the edges, respectively. Both optimizations are constrained and favor idling over detours to reduce mechanical wear. The ILP formulations, tailored to the SaD system, are general enough to be applicable for many other single- and multi-agent problems over discretized networks. We have implemented the ILPs with a gurobi solver. Computational results demonstrate and compare the effectiveness of the two formulations.