Graph-walking automata (GWA) analyze an input graph by moving between its nodes, following the edges. This paper investigates the effect of node-replacement graph homomorphisms on recognizability by these automata. The family of graph languages recognized by GWA is closed under inverse homomorphisms. The main result of this paper is that, for n-state automata operating on graphs with k labels of edge end-points, the inverse homomorphic images require GWA with kn+ O(1 ) states in the worst case. The second result is that already for tree-walking automata, the family they recognize is not closed under injective homomorphisms; here the proof is based on a homomorphic characterization of regular tree languages.