  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.
