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Quasar spectra have a variety of absorption lines whose origins range from energetic winds expelled from the central engines to unrelated, intergalactic clouds. We present multi-epoch, medium resolution spectra of eight quasars at z~2 that have narrow ``associated'' absorption lines (AALs, within $\pm$5000 km s^{-1} of the emission redshift). Two of these quasars were also known previously to have high-velocity mini-broad absorption lines (mini-BALs). We use these data, spanning ~17 years in the observed frame with two to four observations per object, to search for line strength variations as an identifier of absorption that occurs physically near (``intrinsic'' to) the central AGN. Our main results are the following: Two out of the eight quasars with narrow AALs exhibit variable AAL strengths. Two out of two quasars with high-velocity mini-BALs exhibit variable mini-BAL strengths. We also marginally detect variability in a high-velocity narrow absorption line (NAL) system, blueshifted \~32,900 km s^{-1}$ with respect to the emission lines. No other absorption lines in these quasars appeared to vary. The outflow velocities of the variable AALs are 3140 km s^{-1} and 1490 km s^{-1}. The two mini-BALs identify much higher velocity outflows of ~28,400 km s^{-1} and ~52,000 km s^{-1}. Our temporal sampling yields upper limits on the variation time scales from 0.28 to 6.1 years in the quasar rest frames. The corresponding minimum electron densities in the variable absorbers, based on the recombination time scale, are \~40,000 cm^{-3} to ~1900 cm^{-3}. The maximum distances of the absorbers from the continuum source, assuming photoionization with no spectral shielding, range from ~1.8 kpc to ~7 kpc.