Enough electrons will flow between newly-connected plates to bring them to the same potential. From a practical perspective, though, unless something else not shown on the schematic behaves as a significant capacitance connection between the two halves of the circuit, however, the number of electrons that will have to flow to equalize the potentials will be really, really small. Really, really, really, really small.
Think of the voltage at any given point as relating to its willingness to accept or donate electrons; if two points have the same voltage, electrons will have no particular desire to favor one direction or the other. If one point has a higher voltage, electrons will prefer moving in that direction to moving in the reverse. The higher the difference in voltages, the greater the preference.
If an electron arrives on either one of a capacitor's plate, it will make every part of the capacitor be much less accepting of electrons (i.e. decrease its voltage). Conversely, every time an electron leaves one of the plates, it will make every part be much more accepting of electrons (increase the voltage). Note that if many more capacitors enter an electron than leave, both plates will have a very large negative voltage, such that electrons will jump at the opportunity to leave.
What is significant about a capacitor is that when an electron enters one plate, it causes that opposite plate's voltage to decrease slightly less than that of the plate receiving the electron. Consequently, if the balance of electrons entering one plate exceeds that of the other, the plate which receives excess electrons will start to have a more negative voltage than the plate which doesn't, but the amount of imbalance between the two plates that is required to achieve that vastly exceeds the imbalance of electrons entering or leaving the device as a whole which would be required to change both plates' voltage by that amount.
Thus, unless there is some effective capacitance or other connection between the left and right sides, the number of electrons that might have to flow between the two sides to balance out their voltages would be vanishingly insignificant.