The phenomenon of solid-state proton migration within molecular complexes containing short hydrogen bonds is investigated in two dimethylurea-oxalic acid complexes. Extensive characterisation by both X-ray and neutron diffraction shows that proton migration along the hydrogen bond can be induced in these complexes as a function of temperature. This emphasises the subtle features of the hydrogen bond potential well in such short hydrogen bonded complexes, both intrinsically and in the effect of the local crystalline environment. Based on these findings, the synthesis and analysis of a series of solid-state molecular complexes is shown to be a potential route to designing materials with tuneable proton migration effects.