What's the relation between firing of dopaminergic neurons and dopamine dispersion in terms of neurophysiological processes?
I've mostly figured it out now, and I've a better idea of the words and phrases I need to search to get the information I was asking for.

The short answer comes from wikipedia: "The neurons' somata produce the enzymes that synthesize dopamine, and they are then transmitted via the projecting axons to their synaptic destinations, where most of the dopamine is produced." The transmission is a direct result of the firing of the neuron, so DA is dispersed, in the areas where dopaminergic axons end, proportional to the firing of the neurons.

A more complete answer would involve an understanding of the difference between tonic and phasic transmission, for which this has proved an informative and accessible source.

The above sources support the suspicion I mentioned in the hypothetical context provided in the question.