Let consider well-known problem in distributed computing Broadcast Problem in network with stopping crashes.
The most appropriate solution to the broadcast problem is flooding algorithm, when the source of the message just send it to all available nodes of the network on the first round, if there are crashes on the edges some number of message will fail. Then on the second round, the nodes that received messages from the previous round send it to all available nodes and so on.
Let consider slightly different model, when on every round the source node with message is capable to send a message to only one single node of course stopping failure might occur. It's very useful model, which can represent a sort of controllable or limited spreading.
Unfortunately I don't know the formal name of the model. Do you familiar with result in the model of "limited spreading", I am sure there are must some work in this direction.