We examine the effects of stellar feedback and bursty star formation on low-mass galaxies (M star =2×10 6 −5×1010 M e ) using the Feedback in Realistic Environments (FIRE) 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 ∼1 kpc within their first 100 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 ∼200 Myr, and these rapid size fluctuations can account for much of the observed scatter in the radius at fixed M 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 star ≈10 7-9.6 M e , 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 ΛCDM.