  We examine the effects of stellar feedback and bursty star formation on
low-mass galaxies ($M_{\rm star}=2\times10^6-5\times10^{10}{\rm M_{\odot}}$)
using the FIRE (Feedback in Realistic Environments) simulations. While previous
studies emphasized the impact of feedback on dark matter profiles, we
investigate the impact on the stellar component: kinematics, radial migration,
size evolution, and population gradients. Feedback-driven outflows/inflows
drive significant radial stellar migration over both short and long timescales
via two processes: (1) outflowing/infalling gas can remain star-forming,
producing young stars that migrate $\sim1{\rm\,kpc}$ within their first $100
{\rm\,Myr}$, and (2) gas outflows/inflows drive strong fluctuations in the
global potential, transferring energy to all stars. These processes produce
several dramatic effects. First, galaxies' effective radii can fluctuate by
factors of $>2$ over $\sim200 {\rm\,Myr}$, and these rapid size fluctuations
can account for much of the observed scatter in radius at fixed $M_{\rm star}.$
Second, the cumulative effects of many outflow/infall episodes steadily heat
stellar orbits, causing old stars to migrate outward most strongly. This
age-dependent radial migration mixes---and even inverts---intrinsic age and
metallicity gradients. Thus, the galactic-archaeology approach of calculating
radial star-formation histories from stellar populations at $z=0$ can be
severely biased. These effects are strongest at $M_{\rm
star}\approx10^{7-9.6}{\rm M_{\odot}}$, the same regime where feedback most
efficiently cores galaxies. Thus, detailed measurements of stellar kinematics
in low-mass galaxies can strongly constrain feedback models and test baryonic
solutions to small-scale problems in $\Lambda$CDM.
