We present a survey for photometric variability in young, low-mass brown dwarfs with the Spitzer Space Telescope. The 23 objects in our sample show robust signatures of youth and share properties with directly-imaged exoplanets. We present three new young objects: 2MASS J03492367+0635078, 2MASS J09512690 −8023553 and 2MASS J07180871−6415310. We detect variability in 13 young objects, and find that young brown dwarfs are highly likely to display variability across the L2-T4 spectral type range. In contrast, the field dwarf variability occurrence rate drops for spectral types >L9. We examine the variability amplitudes of young objects and find an enhancement in maximum amplitudes compared to field dwarfs. We speculate that the observed range of amplitudes within a spectral type may be influenced by secondary effects such as viewing inclination and/or rotation period. We combine our new rotation periods with the literature to investigate the effects of mass on angular momentum evolution. While high mass brown dwarfs (> 30M Jup ) spin up over time, the same trend is not apparent for lower mass objects (< 30M Jup ), likely due to the small number of measured periods for old, low-mass objects. The rotation periods of companion brown dwarfs and planetary-mass objects are consistent with those of isolated objects with similar ages and masses, suggesting similar angular momentum histories. Within the AB Doradus group, we find a high variability occurrence rate and evidence for common angular momentum evolution. The results are encouraging for future variability searches in directly-imaged exoplanets with facilities such as the James Webb Space Telescope and 30-meter telescopes.