In this paper, we investigate distributed broadcasting in dynamic networks, where the topology changes continually over time. We propose a network model that captures the dynamicity caused by both churn and mobility of nodes. In contrast to existing work on dynamic networks, our model defines the dynamicity in terms of localized topological changes in the vicinity of each node, rather than a global view of the whole network. Obviously, a local dynamic model suits distributed algorithms better than a global one. The proposed dynamic model uses the more realistic SINR model to depict wireless interference, instead of oversimplified graph-based models adopted in most existing work. We consider the fundamental communication primitive of global broadcast, which is to disseminate a message from a source node to the whole network. Specifically, we present a randomized distributed algorithm that can accomplish dynamic broadcasting in an asymptotically optimal running time of $O(D_{T})$ with a high probability guarantee, under the assumption of reasonably constant dynamicity rate, where $D_{T}$ is the dynamic diameter, a parameter proposed to depict the complexity of dynamic broadcasting. We believe our local dynamic model can greatly facilitate distributed algorithm studies in mobile and dynamic wireless networks.