Cold air outbreaks (CAOs) have substantial impacts on public health, energy demand and transportation. Here we identify strong and weak CAO days during boreal winters over 1979-2022 across three major Northern Hemisphere sectors. Strong and weak CAOs occur on average about 11 and 23 days per winter. The 10-day backward trajectories show that air parcels associated with strong CAOs maintain lower potential temperature throughout their evolution, indicating that the strong-weak contrast is established early and persists along the pathway. The trajectories exhibit a common two-stage evolution, with cooling during an intensification stage followed by warming during an outbreak stage as parcels approach the target regions. Decomposition of the peak-intensity contrast, defined as the strong-minus-weak difference in minimum potential temperature, indicates that more than 60% is inherited from differences in the initial thermodynamic state, consistent with more poleward source regions for strong events, while the remainder reflects pathway modification. A Lagrangian dry static energy budget further shows that longwave radiative cooling dominates intensification-stage cooling and is partly offset by turbulent exchange. Strong-weak differences are governed primarily by contrasts in longwave radiation and turbulence. These results provide process-based guidance for improving CAO prediction and risk assessment.