  We present the analysis of multiepoch observations of a set of 12 variable,
Compton-thin, local (z<0.1) active galactic nuclei (AGN) selected from the
100-month BAT catalog. We analyze all available X-ray data from \chandra, \xmm,
and \nustar, adding up to a total of 53 individual observations. This
corresponds to between 3 and 7 observations per source, probing variability
timescales between a few days and $\sim 20$~yr. All sources have at least one
\nustar observation, ensuring high-energy coverage, which allows us to
disentangle the line-of-sight and reflection components in the X-ray spectra.
For each source, we model all available spectra simultaneously, using the
physical torus models \myt, \bor, and \uxc. The simultaneous fitting, along
with the high-energy coverage, allows us to place tight constraints on torus
parameters such as the torus covering factor, inclination angle, and average
column density. We also estimate the line-of-sight column density ($N_{\rm H}$)
for each individual observation. Within the 12 sources, we detect clear
line-of-sight $N_{\rm H}$ variability in 5, non-variability in 5, and for 2 of
them it is not possible to fully disentangle intrinsic-luminosity and $N_{\rm
H}$ variability. We observe large differences between the average values of
line-of-sight $N_{\rm H}$ (or $N_{\rm H}$ of the obscurer) and the average
$N_{\rm H}$ of the torus (or $N_{\rm H}$ of the reflector), for each source, by
a factor between $\sim2$ to $>100$. This behavior, which suggests a physical
disconnect between the absorber and the reflector, is more extreme in sources
that present $N_{\rm H}$ variability. $N_{\rm H}$-variable AGN also tend to
present larger obscuration and broader cloud distributions than their
non-variable counterparts. We observe that large changes in obscuration only
occur at long timescales, and use this to place tentative lower limits on torus
cloud sizes.
