Radiative transfer models of multi-transition observations are used to determine molecular abundances as functions of position in pre- and protostellar cores. The data require a "drop" abundance profile with radius, with high abundances in the outermost regions probed by low excitation 3 mm lines, and much lower abundances at intermediate zones probed by higher frequency lines. The results are illustrated by detailed analysis of CO and HCO ⁺ lines for a subset of objects. We propose a scenario in which the molecules are frozen out in a region of the envelope where the temperature is low enough (≲40 K) to prevent immediate desorption, but where the density is high enough (>10⁴-10⁵ cm^-3) that the freeze-out timescales are shorter than the lifetime of the core. The size of the freeze-out zone is thereby a record of the thermal and dynamical evolution of the cores. Fits to CO data for a sample of 16 objects indicate that the size of the freeze-out zone decreases significantly between class 0 and I objects, explaining the variations in, for example, CO abundances with envelope masses. However, the corresponding timescales are 10^5±0.5 years, with no significant difference between class 0 and I objects. These timescales suggest that the dense pre-stellar phase with heavy depletions lasts only a short time, of the order of 10⁵ yr, in agreement with recent chemical-dynamical models.