  In this work, we study the propagation of influence and computation in
dynamic distributed systems. We focus on broadcasting models under a worst-case
dynamicity assumption which have received much attention recently. We drop for
the first time in worst-case dynamic networks the common instantaneous
connectivity assumption and require a minimal temporal connectivity. Our
temporal connectivity constraint only requires that another causal influence
occurs within every time-window of some given length. We establish that there
are dynamic graphs with always disconnected instances with equivalent temporal
connectivity to those with always connected instances. We present a termination
criterion and also establish the computational equivalence with instantaneous
connectivity networks. We then consider another model of dynamic networks in
which each node has an underlying communication neighborhood and the
requirement is that each node covers its local neighborhood within any
time-window of some given length. We discuss several properties and provide a
protocol for counting, that is for determining the number of nodes in the
network.
