We propose a distributed method for a multi-agent pick-up and delivery problem with fluctuations in agent movement speeds while agents perform planning, detect and resolve conflicts (collisions) between the plans, and execute actions in the plans in a distributed manner. Our study assumes that the robot's movement speed can fluctuate, owing to various factors, thus delaying their scheduled tasks. Such delays can rapidly cause other agent conflicts to cascade and render long-term plans useless. Our proposed method allows each agent's plans to be executed and modified using an advanced short-sighted conflict resolution mechanism. Hence, although an agent attempts to follow its given sequence of actions, it performs each one after carefully checking for any conflict in the next few steps. Our method is fully distributed and works effectively, even when the number of task endpoints, which are the pick-up and delivery locations, is small and the agents are concentrated. We experimentally confirm that our method works efficiently without collisions in environments having agent speed fluctuations and deadlocks using example problems from robot movement in a construction site. Further, we compare the performance of our method with that of the baseline method.