We consider the evolution of strong [Wr(2796)>0.3 Å] Mg II absorbers, most of which are closely related to luminous galaxies. Using 20 high-resolution quasar spectra from the VLT UVES public archive, we examine 33 strong Mg II absorbers in the redshift range 0.3<z<2.5. We compare and supplement this sample with 23 strong Mg II absorbers at 0.4<z<1.4 observed previously with HIRES/Keck. We find that neither equivalent width nor kinematic spread (the optical-depth-weighted second moment of velocity) of Mg II λ2796 evolve. However, the kinematic spread is sensitive to the highest velocity component, and therefore not as sensitive to additional weak components at intermediate velocities relative to the profile center. The fraction of absorbing pixels within the full velocity range of the system does show a trend of decreasing with decreasing redshift. Most high-redshift systems (14/20) exhibit absorption over the entire system velocity range, which differs from the result for low-redshift systems (18/36) at the 95% level. This leads to a smaller number of separate subsystems for high-redshift systems, because weak absorbing components tend to connect the stronger regions of absorption. We hypothesize that low-redshift Mg II profiles are more likely to represent well-formed galaxies, many of which have kinematics consistent with a disk/halo structure. High-redshift Mg II profiles are more likely to show evidence of complex protogalactic structures, with multiple accretion or outflow events. Although these results are derived from measurements of gas kinematics, they are consistent with hierarchical galaxy formation evidenced by deep galaxy surveys.Based on public data obtained from the ESO archive of observations from the UVES spectrograph at the VLT, Paranal, Chile.