We combine dynamical and non-equilibrium chemical modeling of evolving prestellar molecular cloud cores, and explore the evolution of molecular abundances in the contracting core. We model both magnetic cores, with varying degrees of initial magnetic support, and non-magnetic cores, with varying collapse delay times. We explore, through a parameter study, the competing effects of various model parameters in the evolving molecular abundances, including the elemental C/O ratio, the temperature, and the cosmic-ray ionization rate. We find that different models show their largest quantitative differences at the center of the core, whereas the outer layers, which evolve slower, have abundances which are severely degenerate among different dynamical models. There is a large range of possible abundance values for different models at a fixed evolutionary stage (central density), which demonstrates the large potential of chemical differentiation in prestellar cores. However, degeneracies among different models, compounded with uncertainties induced by other model parameters, make it difficult to discriminate among dynamical models. To address these difficulties, we identify abundance ratios between particular molecules, the measurement of which would have maximal potential for discrimination among the different models examined here. In particular, we find that the ratios between NH3 and CO; NH2 and CO; NH3 and HCO+ are sensitive to the evolutionary timescale, and that the ratio between HCN and OH is sensitive to the C/O ratio. Finally, we demonstrate that measurements of the central deviation (central depletion or enhancement) of abundances of certain molecules are good indicators of the dynamics of the core.