  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.
