We study a dynamic version of the Multiple-Message Broadcast problem, where packets are continuously injected in network nodes for dissemination throughout the network. Our performance metric is the ratio of the throughput of such protocol against the optimal one, for any sufficiently long period of time since startup. We present and analyze a dynamic Multiple-Message Broadcast protocol that works under an affectance model, which parameterizes the interference that other nodes introduce in the communication between a given pair of nodes. As an algorithmic tool, we develop an efficient algorithm to schedule a broadcast along a BFS tree under the affectance model. To provide a rigorous and accurate analysis, we define two novel network characteristics based on the network topology, the affectance function and the chosen BFS tree. The combination of these characteristics influence the performance of broadcasting with affectance (modulo a polylogarithmic function). We also carry out simulations of our protocol instantiating affectance in the Radio Network model. To the best of our knowledge, this is the first dynamic Multiple-Message Broadcast protocol that provides throughput guarantees for continuous injection of messages and works under the affectance model.