Multi-agent path finding (MAPF) determines an ensemble of collision-free paths for multiple agents between their respective start and goal locations. Among the available MAPF planners for workspace modeled as a graph, A*-based approaches have been widely investigated due to their guarantees on completeness and solution optimality, and have demonstrated their efficiency in many scenarios. However, almost all of these A*-based methods assume that each agent executes an action concurrently in that all agents start and stop together. This article presents a natural generalization of MAPF with asynchronous actions (MAPF-AA) where agents do not necessarily start and stop concurrently. The main contribution of the work is a proposed approach called Loosely Synchronized Search (LSS) that extends A*-based MAPF planners to handle asynchronous actions. We show LSS is complete and finds an optimal solution if one exists. We also combine LSS with other existing MAPF methods that aims to trade-off optimality for computational efficiency. Numerical results are presented to corroborate the performance of LSS and the applicability of the proposed method is verified in the Robotarium, a remotely accessible swarm robotics research platform.