I will discuss several situations when one has to perform limit passage from non-local to local operators in the context of the degenerate Cahn-Hilliard equation. This includes kinetic derivation of the equation (arXiv:2208.01026, with C. Elbar, M. Mason, B. Perthame), fairly classical problem of passage to the limit from non-local to local equation (arXiv:2208.08955, with C. Elbar) and the same problem for aggregation-diffusion system (in progress, together with J. A. Carrillo, C. Elbar). Not all of these problems are fully understood and to some of them, solutions are available only on the torus.