The McMurdo Dry Valleys magmatic system, Antarctica, provides a world-class example of pervasive lateral magma flow on a continental scale. The lowermost intrusion (Basement Sill) offers detailed sections through the now frozen particle microstructure of a congested magma slurry. We simulated the flow regime in two and three dimensions using numerical models built on a finite-element mesh derived from field data. The model captures the flow behaviour of the Basement Sill magma over a viscosity range of 1–10^{4} Pa s where the higher end (greater than or equal to 10^{2} Pa s) corresponds to a magmatic slurry with crystal fractions varying between 30 and 70%. A novel feature of the model is the discovery of transient, low viscosity (less than or equal to 50 Pa s) high Reynolds number eddies formed along undulating contacts at the floor and roof of the intrusion. Numerical tracing of particle orbits implies crystals trapped in eddies segregate according to their mass density. Recovered shear strain rates (10^{–3}–10^{–5} s^{–1}) at viscosities equating to high particle concentrations (around more than 40%) in the Sill interior point to shear-thinning as an explanation for some types of magmatic layering there. Model transport rates for the Sill magmas imply a maximum emplacement time of *ca* 10^{5} years, consistent with geochemical evidence for long-range lateral flow. It is a theoretically possibility that fast-flowing magma on a continental scale will be susceptible to planetary-scale rotational forces.