We present absorption variability results for 134 bona fide Mg II broad absorption-line (BAL) quasars at 0.46 less than or similar to z less than or similar to 2.3 covering days to similar to 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. Mg II-BAL identifications and related measurements are compiled and presented in a catalog. We find a remarkable time-dependent asymmetry in the equivalent width (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 in high-ionization BALQSO samples, (ii) Mg II-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 Mg II troughs at different velocities in the same quasar, and (iv) the EW-variability correlation between Mg II and Al III BALs is significantly stronger than that between Mg II and C IV 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.