Spreading information through a network of devices is a core activity for most distributed systems. As such, self-stabilizing algorithms implementing information spreading are one of the key building blocks enabling aggregate computing to provide resilient coordination in open complex distributed systems. This paper improves a general spreading block in the aggregate computing literature by making it resilient to network perturbations, establishes its global uniform asymptotic stability and proves that it is ultimately bounded under persistent disturbances. The ultimate bounds depend only on the magnitude of the largest perturbation and the network diameter, and three design parameters trade off competing aspects of performance. For example, as in many dynamical systems, values leading to greater resilience to network perturbations slow convergence and vice versa.