How can I model this usage scenario mathematically?
I think you want a Petri net with transition guards and a global clock.  (Other formalisms would also work, but I'm going with this one because it relates to the discussion on your other question.)

Draw a graph -- vertices and directed edges -- where the vertices are your nodes, and the edges are the communication channels from one node to another.  It sounds as though there's an edge between every node, if all nodes are capable of broadcast, but perhaps I'm misreading.  In any case, the important thing is that you decorate each edge with a rule or set of rules, and the edge only "opens" when the rules are satisfied.  (These rules are called guard expressions.)

For example, to make "vary the amount of time a host caches responses" rigorous, you could make the rule for an edge, "if I receive a message, I wait 3 seconds according to the global clock before I pass it on."  I don't understand your dirty-cache issue well enough to offer anything yet, but I think the conceptual point you need first is this notion of enabling or disabling a transition according to the rules of the system.