Broad absorption lines (BALs) in quasar spectra are prominent signatures of high-velocity outflows, which might be present in all quasars and could be a major contributor to feedback to galaxy evolution. Studying the variability in these BALs allows us to further our understanding of the structure, evolution, and basic physical properties of the outflows. This is the third paper in a series on a monitoring programme of 24 luminous BAL quasars at redshifts 1.2 < z < 2.9. We focus here on the time-scales of variability in CIV 1549A BALs in our full multi-epoch sample, which covers time-scales from 0.02-8.7 yr in the quasar rest-frame. Our sample contains up to 13 epochs of data per quasar, with an average of 7 epochs per quasar. We find that both the incidence and the amplitude of variability are greater across longer time-scales. Part of our monitoring programme specifically targeted half of these BAL quasars at rest-frame time-scales <2 months. This revealed variability down to the shortest time-scales we probe (8-10 days). Observed variations in only portions of BAL troughs or in lines that are optically thick suggest that at least some of these changes are caused by clouds (or some type of outflow substructures) moving across our lines of sight. In this crossing cloud scenario, the variability times constrain both the crossing speeds and the absorber locations. Typical variability times of order ~1 year indicate crossing speeds of a few thousand km/s and radial distances near ~1 pc from the central black hole. However, the most rapid BAL changes occurring in 8-10 days require crossing speeds of 17 000 - 84 000 km/s and radial distances of only 0.001-0.02 pc. These speeds are similar to or greater than the observed radial outflow speeds, and the inferred locations are within the nominal radius of the broad emission line region.
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