We consider a vector reaction-advection-diffusion equation on a hypercube. The measurements are weighted averages of the state over different subdomains. These measurements are asynchronously sampled in time. Subject to matched disturbances, the discrete control signals are applied through shape functions and zero-order holds. The feature of this work is that we consider generalized relay control: the control signals take their values in a finite set. This allows for networked control through low capacity communication channels. First, we derive linear matrix inequalities (LMIs) whose feasibility guarantees the ultimate boundedness with a limit bound proportional to the sampling period. Then we construct a switching procedure for the controller parameters that ensures semi-global practical stability: for an arbitrarily large domain of initial conditions the trajectories converge to a set whose size does not depend on the domain size. For the disturbance-free system this procedure guarantees exponential convergence to the origin. The results are demonstrated by two examples: 2D catalytic slab and a chemical reactor. (C) 2017 Elsevier Ltd. All rights reserved.