We propose a distributed planning method with asynchronous execution for multi-agent pickup and delivery (MAPD) problems for environments with occasional delays in agents' activities and flexible endpoints. MAPD is a crucial problem framework with many applications; however, most existing studies assume ideal agent behaviors and environments, such as a fixed speed of agents, synchronized movements, and a well-designed environment with many short detours for multiple agents to perform tasks easily. However, such an environment is often infeasible; for example, the moving speed of agents may be affected by weather and floor conditions and is often prone to delays. The proposed method can relax some infeasible conditions to apply MAPD in more realistic environments by allowing fluctuated speed in agents' actions and flexible working locations (endpoints). Our experiments showed that our method enables agents to perform MAPD in such an environment efficiently, compared to the baseline methods. We also analyzed the behaviors of agents using our method and discuss the limitations.