Random network models have been a popular tool for investigating cortical network dynamics. On the scale of roughly a cubic millimeter of cortex, containing about 100,000 neurons, cortical anatomy suggests a more realistic architecture. In this locally connected random network, the connection probability decreases in a Gaussian fashion with the distance between neurons. Here we present three main results from a simulation study of the activity dynamics in such networks. First, for a broad range of parameters these dynamics exhibit a stationary state of asynchronous network activity with irregular single-neuron spiking. This state can be used as a realistic model of ongoing network activity. Parametric dependence of this state and the nature of the network dynamics in other regimes are described. Second, a synchronous excitatory stimulus to a fraction of the neurons results in a strong activity response that easily dominates the network dynamics. And third, due to that activity response an embedding of a divergent-convergent feed-forward subnetwork (as in synfire chains) does not naturally lead to a stable propagation of synchronous activity in the subnetwork; this is in contrast to our earlier findings in isolated subnetworks of that type. Possible mechanisms for stabilizing the interplay of volleys of synchronous spikes and network dynamics by specific learning rules or generalizations of the subnetworks are discussed.