In my experience, category theory is very successful at providing powerful machinery to reason about large objects or objects unrestricted in size, for example (logical) models (via accessible categories) or "nice" topological spaces (via simplicial sets). However I haven't seen many impressive applications of category theory to categories of finite objects, for example finite graphs or more generally finite models, whose behaviors can be radically different from the corresponding objects of unrestricted size. I certainly haven't read widely enough, but I want to ask anyway: is this finite/infinite divide a conception or misconception? If the former, are there any intuitive explanations? If the latter, what are some convincing counterexamples (for example, applications that provided major insight in what graph theorists would be interested in)?