Max network flow with arbitrary source / sink
Finding max flow from arbitrary source/sink can be reduced to finding max flow under insertion/deletion of edges.

We modify the graph by adding two vertices $s^*$ and $t^*$ not connected to any vertex. We maintain the data structure to compute max flow from $s^*$ to $t^*$ under insertion/deletion of edges. Now, whenever we want to calculate max flow from any source $s'$ to any sink $t'$, we simply add edges $(s^*,s')$ and $(t',t^*)$ with infinite capacity. The max flow from $s^*$ to $t^*$ would then essentially be the max flow from $s'$ to $t'$. 

In case you want to change the source and sink to $s''$ and $t''$ respectively, simply delete $(s^*,s')$ and $(t',t^*)$ and add $(s^*,s'')$ and $(t'',t^*)$. The max flow from $s^*$ to $t^*$ would now represent the max flow from $s''$ to $t''$.

This technique was earlier used to show the equivalence of All Pair Shortest Path and Single Source Shortest Path in the dynamic environment.
Similarly, it can also show equivalence of All Pair Reachability and Single Source Reachability in the dynamic environment.