We present 13CO and C18O (1-0), (2-1), and (3-2) maps towards the core-forming Perseus B1-E clump using observations from the James Clerk Maxwell Telescope (JCMT), Submillimeter Telescope (SMT) of the Arizona Radio Observatory, and IRAM 30 m telescope. We find that the 13CO and C18O line emission both have very complex velocity structures, indicative of multiple velocity components within the ambient gas. The (1-0) transitions reveal a radial velocity gradient across B1-E of 1 km/s/pc that increases from north-west to south-east, whereas the majority of the Perseus cloud has a radial velocity gradient increasing from south-west to north-east. In contrast, we see no evidence of a velocity gradient associated with the denser Herschel-identified substructures in B1-E. Additionally, the denser substructures have much lower systemic motions than the ambient clump material, which indicates that they are likely decoupled from the large-scale gas. Nevertheless, these substructures themselves have broad line widths (0.4 km/s) similar to that of the C18O gas in the clump, which suggests they inherited their kinematic properties from the larger-scale, moderately dense gas. Finally, we find evidence of C18O depletion only toward one substructure, B1-E2, which is also the only object with narrow (transonic) line widths. We suggest that as prestellar cores form, their chemical and kinematic properties are linked in evolution, such that these objects must first dissipate their turbulence before they deplete in CO. ; Comment: Accepted by ApJ, 34 pages, 12 figures