In simple pseudo-time-dependent models of the gas-phase chemistry of dense interstellar clouds in which there is more elemental oxygen than carbon, the calculated abundances of organic molecules peak at a so-called "early time" of 1-3 x 10(5) yr before decreasing markedly to steady state values. The large observed abundances of these molecules in dark clouds such as TMC-1 as well as in extended regions of the cores of giant clouds are accounted for in pseudo-time-dependent models only at or near "early time." Since the actual ages of dense interstellar clouds are much larger than 10(5) yr, it is desirable that a mechanism be elucidated which preserves the calculated early-time abundances for much longer periods of time. One such mechanism stems from the phenomenon of dynamical mixing of gaseous material between diffuse regions and denser portions of interstellar clouds. As originally discussed by Chieze and Pineau des Forets, such dynamical mixing can continually transfer large amounts of neutral and singly ionized atomic carbon into the dense cloud gas; these species, normally present in large abundance only at or before "early time" in simple one-phase models, enhance the synthesis of organic molecules. The possibility thus exists that mixing can lead to large calculated abundances of organic molecules in dense gas at times much later than 10(5) yr and even obliterate the distinction between early time and steady state. In this paper, we present model calculations which show the situation to be somewhat more complex than this. If one considers mixing between diffuse and dense clumps with no gas intermediate in density, it is necessary to mix in large amounts of neutral atomic carbon rather than the singly ionized variety to maintain high abundances of organic molecules at t >> 10(5) yr. If, on the other hand, one considers a more natural model in which gas condensation occurs before mixing, the very act of compression is a significant source of complex molecules. Furthermore, high abundances of organic molecules are preserved for a large range of mixing times.