Could some of it have to do with the difference between say considering very general structures versus working in a specific structure? For example, if one is trying to prove existence of a solution (in some sense) to PDEs, one may need to work on finding a priori bounds for some specific PDEs. The techniques used to find these bounds may or may not be able to be generalized (for example, many PDEs of interest have non-linear terms of differing "flavors"). Then on the other hand, is searching for a priori bounds alone something that the existence theories of these different PDEs have in common, if you will?

As far as "global reasoning," this type of reasoning would seem to be the correct approach when dealing with general structures (examples: Banach Spaces, Algebras, &c). If one is interested in proving statements about structures without imposing too many assumptions, then it would seem that thinking "globally" might be beneficial.