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Traditional task assignment approaches for multi-agent motion control do not take the possibility of collisions into account. This can lead to challenging requirements for path planning. We derive an assignment method that not only minimises the largest distance between an agent and its assigned destination but also provides local constraints for guaranteed collision avoidance. To this end, we introduce a sequential bottleneck optimisation problem and define a notion of robustness of an optimising assignment to changes of individual assignment costs. Conditioned on a sufficient level of robustness in relation to the size of the agents, we construct time-varying position bounds for every individual agent. These local constraints are a direct byproduct of the assignment procedure and only depend on the initial agent positions, the destinations that are to be visited, and a timing parameter. We prove that no agent that is assigned to move to one of the target locations collides with any other agent if all agents satisfy their local position constraints. We demonstrate the method in a illustrative case study.