  [Abridged] We present 23 quasar absorption line systems selected by the MgII
doublet with W_r(2796)>0.3 Ang over the redshift range 0.4<z<1.2. The
kinematics are studied at ~6 km/s resolution to a 5-sigma detection threshold
of 0.015 Ang. Most absorbers are characterized by a dominant kinematic
subsystem with W_r>0.2 Ang and velocity spreads ranging from 10-50 km/s, in
proportion to the system equivalent width. Additional kinematic subsystems have
velocities out to ~400 km/s. The equivalent widths and velocity spreads of
these "outlying", weaker subsystems are anti-correlated with their velocities
and their equivalent width distribution turns down from a power law below
W_r~0.08 Ang. These "moderate" and "high velocity" subsystems and probably not
higher redshift analogues to Galactic high velocity clouds (HVCs). Weak
subsystems are asymmetrically distributed in velocity; they are either all
blueshifted or all redshifted with respect to the dominant subsystem. This
implies, that on a case-by-case basis, a given line of sight is apparently
probing a well defined spatial and kinematic structure. We investigate a simple
kinematic model that relies on a rotating disk to explain the observed
asymmetries. There are systematic differences, or trends, in both the subsystem
to subsystem velocity clustering and in the overall kinematics with increasing
equivalent width; we discuss how these may provide clues to the observed
differential evolution in the equivalent width distribution of MgII absorbers.
