In this work, we focus on iterative distributed model predictive control (DMPC) of large-scale nonlinear systems subject to asynchronous, delayed state feedback. The motivation for studying this control problem is the presence of asynchronous, delayed measurement samplings in chemical processes and the potential use of networked sensors and actuators in industrial process control applications to improve closed-loop performance. Under the assumption that there exist upper bounds on the time interval between two successive state measurements and on the maximum measurement delay, we design an iterative DMPC scheme for nonlinear systems via Lyapunov-based control techniques. Sufficient conditions under which the proposed distributed MPC design guarantees that the state of the closed-loop system is ultimately bounded in a region that contains the origin are provided. The theoretical results are illustrated through a catalytic alkylation of benzene process example.