  Spatial patterns of stellar elemental abundances encode rich information
about a galaxy's formation history. We analyze the radial, vertical, and
azimuthal variations of metals in stars, both today and at formation, in the
FIRE-2 cosmological simulations of Milky Way (MW)-mass galaxies, and we compare
with the MW. The radial gradient today is steeper (more negative) for younger
stars, which agrees with the MW, although radial gradients are shallower in
FIRE-2. Importantly, this age dependence was present already at birth: radial
gradients today are only modestly ($\lesssim$ 0.01 dex kpc$^{-1}$) shallower
than at birth. Disk vertical settling gives rise to negative vertical gradients
across all stars, but vertical gradients of mono-age stellar populations are
weak. Similar to the MW, vertical gradients in FIRE-2 are shallower at larger
radii, but they are overall shallower in FIRE-2. This vertical dependence was
present already at birth: vertical gradients today are only modestly
($\lesssim$ 0.1 dex kpc$^{-1}$) shallower than at birth. Azimuthal scatter is
nearly constant with radius, and it is nearly constant with age $\lesssim$ 8
Gyr ago, but increases for older stars. Azimuthal scatter is slightly larger
($\lesssim$ 0.04 dex) today than at formation. Galaxies with larger azimuthal
scatter have a stronger radial gradient, implying that azimuthal scatter today
arises primarily from radial redistribution of gas and stars. Overall, spatial
variations of stellar metallicities show only modest differences between
formation and today; spatial variations today primarily reflect the conditions
of stars at birth, with spatial redistribution of stars after birth
contributing secondarily.
