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Proton-transfer (PT) between organic complexes is a common and important biochemical process. Unfortunately, PT energy barriers are difficult to accurately predict using density functional theory (DFT); in particular, the generalized gradient approximation (GGA) tends to underestimate PT barriers. Moreover, PT typically occurs in environments where dispersion forces contribute to the cohesion of the system; thus, a suitable exchange-correlation functional should accurately describe both dispersion forces and PT barriers. This paper provides benchmark results for the PT barriers of several density functionals including several variants of the van der Waals density functional (vdWDF). The benchmark set comprises small organic molecules with inter- and intra-molecular PT. The results show that replacing GGA correlation with a fully non-local vdW-DF correlation increases the PT barriers, making it closer to the quantum chemical reference values. In contrast, including nonlocal correlations with the Vydrov-Voorhis (VV) method or dispersion-corrections at the DFT-D3 or the Tkatchenko-Scheffler (TS) level has barely any impact on the PT barriers. Hybrid functionals also increase and improve the energies and the best performance is provided by a hybrid version of the consistent-exchange van der Waals density functional vdW-DF-cx. For the formic acid dimer PT system, we analyzed the GGA exchange and non-local correlation contributions. The analysis shows that the repulsive part of the non-local correlation kernel plays a key role in the PT energy barriers predicted with vdW-DF.