Given a network flow, are there bounds on the change in weight on nodes?
I think you're using a confusing sign convention, but I'll stick with it.  It's pretty easy to see that for any connected graph you can have all weight flowing into a single vertex (unless I'm misunderstanding something), so the lower bound you'll get is $$\Delta_i\geq -\sum_{i\in V}w_i.$$

Things won't really be better for bounding the square.  For example, if your graph is a star in which every leaf has $w_i=1$, then $$\sum_{i\in V}\Delta_i^2 = (|V|-1)+(|V|-1)^2.$$

You can't get a neighbourhood restriction, either, because you can take the example of a $k$-ary tree in which every leaf has weight 1 and all the weight goes to the root.

What you can get, however, is the possibly useful bound $$\sum_{i\in V}{|\Delta_i|} \leq 2\sum_{i\in V}w_i.$$

To see this, just note that when charge leaves a vertex you'll count it twice.