Is there such thing as "half-life" of dopamine?
Dopamine (DA) secreted by neurons is selectively reuptaken by the dopamine transporter (DAT), present on nerve terminals.


(source: inist.fr)  

Image source: Plasma membrane monoamine transporters: structure, regulation and function - Torres et al. - Nat Rev Neurosci., 2003

DAT acts as a symporter, using the Na+ and Cl- gradients to generate the energy required to transport DA back in the cytosol.

The mechanism of reuptake has been investigated in vitro allowing the description of a model for DA reuptake by DAT.

In particular, from the abstract of this paper:
A multisubstrate mechanism of striatal dopamine uptake and its inhibition by cocaine. - McElvain and Schenk, Biochem Pharmacol., 1992


  Dopamine uptake was found to be first order in dopamine with a Vmax of 582 pmol/sec/g wet weight and a Km of 1.2 μM. The results of experiments in which choline and isethionate were substituted for Na+ and Cl−, respectively, suggested that the uptake process is second order in Na+ and first order in Cl−. Multisubstrate analyses of the initial velocities of uptake over the concentration range of 0.025 to 1.5 μM dopamine suggested that the mechanism of binding of dopamine to the uptake carrier is a partially random, sequential mechanism where dopamine or Na+ binds first with the uptake carrier and Cl− binds last.


A transport scheme is reported in:
Dopamine Neuronal Transport Kinetics and Effects of Amphetamine




  Scheme 1 is a kinetic scheme for a simple carrier adapted from that of Stein (1986) for the DAT. The subscripts o and i represent the outside and inside, respectively, of a DA neuron. Scheme 2 describes a possible mechanism for
  the cotransport of another species. All symbols are as in scheme 1 except that A corresponds to amphetamine.


From the text


  This scheme describes transport as a series of steps: the competition between binding and dissociation of the substrate to the transporter on the  extracellular side, a similar process on the cytoplasmic side, and an equilibrium between the unoccupied transporter facing toward the extracellular or cytoplasmic side. The driving force for inward transport arises from the cotransport of Na1 and Cl2 as well as the membrane potential difference
  (Stein, 1986; Sonders et al., 1997). Biochemical (Krueger, 1990) and electrophysiological (Sonders et al.,1997) studies have established that DA transport is accompanied by transport of at least two Na1 ions and one Cl2 ion and that DA transport can be bidirectional in isolated preparations.