We present a mechanism of topological pumping in origami metamaterials with spatial modulation by tuning the rotation angles. Through coupling spatially modulated origami chains along an additional synthetic dimension, the pumping of waves from one topological edge state to another is achieved, where the Landau-Zener transition is demonstrated by varying the number of coupled origami chains. In addition, the inherent nonlinearity of origami metamaterials enables the excitation-dependent Landau-Zener tunneling probability. Furthermore, with the increase of nonlinearity, the topological states tend to localize in several regions in a way reminiscent of discrete breathers. Our findings pave the way towards interband transitions and associated topological pumping features in origami metamaterials.