The distribution of gas-phase abundances in galaxy disks encodes the history of nucleosynthesis and transport through the interstellar medium (ISM) over cosmic time. Multi-object and high resolution integral-field spectroscopy have started to measure these distributions across hundreds of HII regions individually resolved at $lesssim 100$ pc scales in a handful of objects, but in the coming decade these studies will expand to larger samples of galaxies. This will allow us to understand the role of feedback and turbulence in driving the mixing and diffusion of metals in the ISM, and statistically assess the role of galaxy environment and disk dynamics in modifying how mixing proceeds. Detailed searches for over- and under-enriched regions can address to what extent star formation is triggered by previous generations of star formation and by pristine and recycled gas flows. Local galaxies, for which these detailed measurements will be possible within the next decade, will inform the interpretation of integrated measurements at high-z, where very different dynamical gas-rich environments are found in early disk galaxies. Currently, progress in the field is severely hampered by the 0.2-0.3 dex level systematic uncertainties plaguing nebular abundance diagnostics. Improving our detailed understanding of ionized nebulae at $<$20 pc scales will help us find a solution to this problem, which will prove key to the study of metal enrichment and mixing across the galaxy population in the next decade.
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