Local hybrid functionals with position-dependent exact-exchange admixture are a new class of exchange-correlation functionals in density functional theory that promise to advance the available accuracy in many areas of application. Local hybrids with different local mixing functions (LMFs) governing the position dependence are validated for the heats of formation of the extended G3/99 set, and for two sets of barriers of hydrogen-transfer and heavy-atom transfer reactions (HTBH38 and NHTBH38 databases). A simple local hybrid Lh-SVWN with only Slater and exact exchange plus local correlation and a one-parameter LMF, g(r)=b(tau(W)(r)tau(r)), performs best and provides overall mean absolute errors for thermochemistry and kinetics that are a significant improvement over standard state-of-the-art global hybrid functionals. In particular, this local hybrid functional does not suffer from the systematic deterioration that standard functionals exhibit for larger molecules. In contrast, local hybrids based on generalized gradient approximation exchange tend to give rise to nonintuitive LMFs, and no improved functionals have been obtained along this route. The LMF is a real-space function and thus can be analyzed in detail. We use, in particular, graphical analyses to rationalize the performance of different local hybrids for thermochemistry and reaction barriers.