We consider locally checkable labeling LCL problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between $ω(\log^* n)$ and $o(\log n)$. This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. 1. We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where $O(1)$, $Θ(\log^* n)$, and $Θ(n)$ are the only possible complexity classes. 2. It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the "only if" part of the conjecture. 3. In addition to the well-known complexity landscapes for paths, trees, and general graphs, there are infinitely many different complexity landscapes among minor-closed graph classes.