  We have traced the spatial distributions of intermediate-age and old stars in
nine dwarf galaxies in the distant parts of the Local Group, using multi-epoch
3.6 and 4.5 micron data from the DUST in Nearby Galaxies with Spitzer
(DUSTiNGS) survey. Using complementary optical imaging from the Hubble Space
Telescope, we identify the tip of the red giant branch (TRGB) in the 3.6 micron
photometry, separating thermally-pulsating asymptotic giant branch (TP-AGB)
stars from the larger red giant branch (RGB) populations. Unlike the constant
TRGB in the I-band, at 3.6 micron the TRGB magnitude varies by ~0.7 mag, making
it unreliable as a distance indicator. The intermediate-age and old stars are
well mixed in two-thirds of the sample with no evidence of a gradient in the
ratio of the intermediate-age to old stellar populations outside the central
~1-2'. Variable AGB stars are detected in the outer extremities of the
galaxies, indicating that chemical enrichment from these dust-producing stars
may occur in the outer regions of galaxies with some frequency. Theories of
structure formation in dwarf galaxies must account for the lack of radial
gradients in intermediate-age populations and the presence of these stars in
the outer extremities of dwarfs. Finally, we identify unique features in
individual galaxies, such as extended tidal features in Sex A and Sag DIG and a
central concentration of AGB stars in the inner regions of NGC 185 and NGC 147.
