The abundance dispersion of metal-poor stars in the halo of the Milky Way (MW)suggests that the interstellar medium (ISM) was not chemically well-mixed at earlytimes. The degree to which turbulence, driven by core collapse supernovae (cc-SNe),mixes the metals produced in those same events should provide constraints on the localstar formation conditions at the time of formation of the metal-poor stars. Furthermore,the difference in scatter observed for different elements, particularly α- and r-processelements, holds important clues to the formation processes of these elements.The focus of this thesis has been to use a series of simulations of gas dynamics in galacticdisks to isolate the processes by which cc-SNe drive turbulent mixing in the gas, withthe aim of understanding how the mixing process imprints on the abundance dispersionof the gas. By resolving the metal mixing effects on parsec scales, in set-ups that mimicdifferent galaxy types it is possible to measure the turbulent diffusion coefficient indifferent environments. The diffusion coefficient is an important input parameter forlarger simulations which are currently forced to model mixing at a sub-grid level.This work shows that it is possible to use the star-to-star abundance dispersion ofα-elements to derive constraints for the star formation rate (SFR) of the birth environ-ment of the metal-poor stars, and that this spread is compatible with star formationconditions in a dwarf or a low SFR MW progenitor.This new insight into the relationship between turbulent mixing and abundance dis-persion is subsequently applied to understanding the shape of the r-process elementdistribution. By exploring a large parameter space of rates and masses per event it isshown that the relative rate of r-process events to cc-SNe directly informs the shape ofthe element distribution. This is a useful result for future endeavours to constrain the astrophysical site(s) of r-process production.The observed star-to-star dispersion in r-process abundances can be used to constrainthe the relative rate of r-process enrichment which in turn places new constraints on theexpected total production rate of r-process elements in the MW. The new constraintsare compatible with previous, independent measurements and constitute a significantreduction in the allowed parameter space.Finally, the galaxy patch simulations done in this thesis are well suited to understandingthe fraction of metals that are lost from galaxy systems of different masses. It is foundthat smaller systems, e.g. dwarf galaxies, should loose a significant fraction of newmetals (both α and r-process elements), specifically for r-process enrichment this resultssuggests that smaller systems may require a higher rate of enrichment to explain theobserved abundances.