What happens to current flow in a capacitive circuit when the DC voltage across the capacitor is approximately equal to the source voltage?

The current becomes zero as soon as the difference in voltage of both sides of the capacitor plates becomes the same as the voltage of the source voltage.

If the switch in the circuit  is opened, the plates remain charged. Otherwise , the capacitor quickly discharges when it is short circuite

capacitor is fully charged at this state it looks for discharge path .  so current through capacitor is zero.