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Broad absorption lines (BALs) in quasar spectra indicate high-velocity outflows that may be present in all quasars and could be an important contributor to feedback to their host galaxies. Variability studies of BALs help illuminate the structure, evolution and basic physical properties of the outflows. Here we present further results from an ongoing BAL monitoring campaign of a sample of 24 luminous quasars at redshifts 1.2 < z < 2.9. We directly compare the variabilities in the C IVλ1549 and Si IVλ1400 absorption to try to ascertain the cause(s) of the variability. We find that Si IV BALs are more likely to vary than C IV BALs. When looking at flow speeds >-20 000 km/s, 47 per cent of quasars exhibited Si IV variability while 31 per cent exhibited C IV variability. Furthermore, ∼50 per cent of the variable Si IV regions did not have corresponding C IV variability at the same velocities, while nearly all occurrences of C IV variability had corresponding changes in Si IV. We do not find any correlation between the absolute change in strength in C IV and in Si IV, but the fractional change in strength tends to be greater in Si IV than in C IV. When both C IV and Si IV varied, those changes always occurred in the same sense (either getting weaker or stronger). We also include our full data set so far in this paper, which includes up to 10 epochs of data per quasar. The multi-epoch data show that the BAL changes were not generally monotonic across the full ∼5-8 yr time span of our observations, suggesting that the characteristic time-scale for significant line variations, and (perhaps) for structural changes in the outflows, is less than a few years. Coordinated variabilities between absorption regions at different velocities in individual quasars seem to favour changing ionization of the outflowing gas as the cause of the observed BAL variability. However, variability in limited portions of broad troughs fits naturally in a scenario where movements of individual clouds, or substructures in the flow, across our lines of sight cause the absorption to vary. The actual situation may be a complex mixture of changing ionization and cloud movements. Further discussion of the implications of variability, e.g. in terms of the size and location of the outflowing gas, will be presented in a forthcoming paper.