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 analyzed all available X-ray data from Chandra, XMM-Newton, and NuSTAR, adding up to a total of 53 individual observations. This corresponds to between three and seven observations per source, probing variability timescales between a few days and ∼20 yr. All sources have at least one NuSTAR observation, ensuring high-energy coverage, which allowed us to disentangle the line-of-sight and reflection components in the X-ray spectra. For each source, we modeled all available spectra simultaneously, using the physical torus models MYTorus, borus02, and UXCLUMPY. The simultaneous fitting, along with the high-energy coverage, allowed us to place tight constraints on torus parameters such as the torus covering factor, inclination angle, and torus average column density. We also estimated the line-of-sight column density (NH) for each individual observation. Within the 12 sources, we detected clear line-of-sight NH variability in five of them, non-variability in five of them, and for two of them it was not possible to fully disentangle intrinsic luminosity and NH variability. We observed large differences between the average values of line-of-sight NH (or NH of the obscurer) and the average NH of the torus (or NH of the reflector), for each source, by a factor between ∼2 to > 100. This behavior, which suggests a physical disconnect between the absorber and the reflector, is more extreme in sources that present NH variability. We note that NH-variable AGN also tend to present larger obscuration and broader cloud distributions than their non-variable counterparts. These trends however require a larger number of sources to confirm (or disprove) this. We observed that large changes in obscuration only occur at long timescales, and used this to place tentative lower limits on torus cloud sizes. Furthermore, we observed a median variation in NH between any two observations of the same source of ∼36%.
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