Hydrogen is attracting increasing attention as a clean-burning, carbon-neutral fuel in heavy-duty engine applications, which usually often operate by compression ignition. However, hydrogen cannot ignite on its own under typical conditions in these engines. To address this, the concept of hydrogen-diesel dual direct injection (H₂DDI) has been recently proposed in which hydrogen and diesel are introduced separately into the combustion chamber, with diesel providing the ignition source, leading to ignition occurring in stratified mixtures. This study employs numerical simulations to investigate the ignition of stratified mixtures in one-dimensional laminar mixing-layer configurations under engine-relevant conditions, where a n-dodecane fuel-rich stream is surrounded by an oxidiser stream with (dual-fuel/DF cases) and without (single fuel/SF cases) hydrogen. It is shown that the ignition of the DF cases is delayed compared to the SF cases because hydrogen consumes OH species during the low-temperature oxidation of n-dodecane. Unlike the SF cases, the DF cases exhibit a postponed initiation of the first-stage ignition relative to the most reactive ignition delay time observed in homogeneous reactor simulations. In both the SF and DF cases, diffusion-supported cool flames promote radical accumulation, accelerating the transition to the second-stage ignition. Furthermore, the study reveals that preferential diffusion effects in both the SF and DF cases, influenced by mixing conditions, significantly impact the ignition process, with the diffusion behaviours of different species playing distinct roles in the ignition of both stages. In particular, the preferential diffusion of n-dodecane is the dominant factor of overall preferential diffusion effects, delaying both the first and second-stage ignition, whereas the preferential diffusion of hydrogen only increases the maximum flame temperature.