You can find it in the paper: "Unknotting number, genus, and companion tori" by Scharlemann and Thompson MR0929535

They use sutured manifold theory to show that if a crossing change reduces the genus of a satellite knot by at least 2 then the loop along which Dehn surgery is done to effect the crossing change can be isotoped to be disjoint from each essential torus.

Off the top of my head, I don't see a way to convert this into a statement about diagrams.