This study investigates the interaction of compositional effects with the flow behaviour during near-miscible (and immiscible) CO2-oil displacements in heterogeneous systems. A series of numerical simulations modelling 1D slim-tube and 2D areal systems were simulated using a fully compositional simulator. A number of grid resolutions for a slim-tube model were simulated to choose the proper level of numerical dispersion to mimic the actual physical dispersion. The corresponding 2D cases are based on a small heterogeneous sector model of dimensions 50 m x 10 m, in order that the fine-scale displacement physics can be modelled accurately. We investigated various flow regimes ranging from viscous fingering to channelling displacements within heterogeneous random correlated fields. We found that the reduced recovery is the result of a combination of differences in sweep efficiency associated with the viscous fingering and possible differences in local mixing that affect composition path. At the same time, the unstable phase flow determined by the underlying heterogeneity slows the flow in the unswept area and leads to unequal displacement performance between preferential and non-preferential routes. Specifically, lighter components have moved preferentially in high gas saturation zones, and leaving the heavier components behind in slower flow zones. In the case of channelling flow, compositional effects were less important since the permeability channel dominated the displacement. Both the ultimate oil recovery and component recovery are significantly and about equally reduced, when the underlying heterogeneity is of dominant influence. To summarise, compositional effects can have a very significant impact on the prediction of near-miscible CO2 EOR projects. Issues such as front stability, local displacement efficiency and formation of fingering/channelling during CO2 near-miscible displacement can lead to behaviour that is significantly different from immiscible flooding in these systems. The process of mass transfer between CO2 and oil can be hampered to a certain degree by unstable flow depending on the level of heterogeneity. This leads to a further reduction in component recovery, particularly of the heavier components. The complete dataset and results of this study are available online as a model case example for compositional flows in heterogeneous systems (Wang et al. in "The analysis of compositional effects on global flow regimes in CO2 near-miscible displacements in heterogeneous systems" dataset for paper SPE-190273, 2018. 10.17861/fc1c90bb-9d3f-4a6c-9170-7b7fe10ec7b9).