In this work, we address distributed model predictive control of nonlinear process systems subject to asynchronous measurements. Asynchronous measurements arise naturally in process control applications where, for example, species concentrations or particle size distributions are measured. Assuming that there exists an upper bound on the interval between two successive measurements of the process state, two separate Lyapunov-based model predictive controllers that coordinate their actions and take asynchronous measurements explicitly into account are designed. Sufficient conditions under which the proposed distributed control design guarantees that the state of the closed-loop system is ultimately bounded in a region that contains the origin are provided. In addition, the proposed distributed control design only requires one directional communication between the two distributed controllers and provides the potential of maintaining stability and performance in the face of new or failing actuators. The results are illustrated through a chemical process example.