  We present absorption variability results for 134 bona fide \mgii\ broad
absorption line (BAL) quasars at 0.46~$\lesssim z \lesssim$~2.3 covering days
to $\sim$ 10 yr in the rest frame. We use multiple-epoch spectra from the Sloan
Digital Sky Survey, which has delivered the largest such BAL-variability sample
ever studied. \mgii-BAL identifications and related measurements are compiled
and presented in a catalog. We find a remarkable time-dependent asymmetry in EW
variation from the sample, such that weakening troughs outnumber strengthening
troughs, the first report of such a phenomenon in BAL variability. Our
investigations of the sample further reveal that (i) the frequency of BAL
variability is significantly lower (typically by a factor of 2) than that from
high-ionization BALQSO samples; (ii) \mgii\ BAL absorbers tend to have
relatively high optical depths and small covering factors along our line of
sight; (iii) there is no significant EW-variability correlation between \mgii\
troughs at different velocities in the same quasar; and (iv) the EW-variability
correlation between \mgii\ and \aliii\ BALs is significantly stronger than that
between \mgii\ and \civ\ BALs at the same velocities. These observational
results can be explained by a combined transverse-motion/ionization-change
scenario, where transverse motions likely dominate the strengthening BALs while
ionization changes and/or other mechanisms dominate the weakening BALs.
