Flow of electrons when two charged plates are connected by wire
Consider an analogy
Consider two water tanks with first tank having a higher water level in it. These tanks are connected to each other as shown in figure.



Case 1: The interconnect is a water filled hollow pipe: The water will flow from left tank to right tank.

Case 2: The interconnect is a hollow pipe filled with ice: There won't be any water flow.

In both cases, the pipes are filled with same molecules (\$H_2O\$). But the flow will happen only if the molecules are free to flow. Bounded molecules (as in ice), won't contribute to flow. 

Electrons are like these water molecules. They will contribute to flow (current) only if they are mobile. So the excess electrons from negative plate to the positive plate will flow only if the interconnect have free (mobile) electrons in it.