We initiate the program of extending to higher-rank graphs (k-graphs) the geometric classification of directed graph -algebras, as completed in Eilers et al. (2016, Preprint). To be precise, we identify four moves, or modifications, one can perform on a k-graph, which leave invariant the Morita equivalence class of its -algebra. These moves - in-splitting, delay, sink deletion, and reduction - are inspired by the moves for directed graphs described by Sorensen (Ergodic Th. Dyn. Syst. 33(2013), 1199-1220) and Bates and Pask (Ergodic Th. Dyn. Syst. 24(2004), 367-382). Because of this, our perspective on k-graphs focuses on the underlying directed graph. We consequently include two new results, Theorem 2.3 and Lemma 2.9, about the relationship between a k-graph and its underlying directed graph.