The inclusion of molecular diffusion into the joint-probability density function (JPDF) method requires terms for molecular transport in physical space and mixing in composition space. The former is typically neglected for high Reynolds number flows but becomes important in flows close to surfaces and for flamelet related reaction-diffusion structures associated with high Damköhler numbers. McDermott and Pope (J. Comp. Phys., 2007) proposed an explicit addition of a molecular transport term to the equation for sub-grid molecular diffusion for the filtered density function method. A related approach was used by Fiolitakis et al. (Combust. Flame, 2014) for the JPDF method when combined with moment based closures. A variant of the method is here combined with comprehensive C1−C2 chemistry featuring 44 chemical species and 256 reactions to study the impact on auto-ignition and flame stabilisation in the well-characterised Cabra burner. It is shown that the contribution of the molecular transport term becomes significant at the flame stabilisation point with predictions of carbon monoxide and, in particular, molecular hydrogen showing a marked improvement. Conditional statistics and the PDF of temperature show that the improvement is due to increased molecular diffusion on the fuel rich side of the flame leading to increased chemical activity. The developed approach is viable for boundary layers while the spatial resolution requirements in terms of the mean scalar gradients may prove limiting for complex flows.