  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 ($>30
M_{\mathrm{Jup}}$) spin up over time, the same trend is not apparent for lower
mass objects ($<30 M_{\mathrm{Jup}}$), 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.
