When an AC source is connected to a capacitor there is a steady-state current in the circuit. Does it mean that the charges jump from one plate to the other to complete the circuit?
No.
When an AC source is connected to a capacitor, there is a steady-state current in the circuit which transfers charges smoothly between the plates of the capacitor. This results in a potential difference across the plates of the capacitor. The direction of current is alternatively reversed every half-cycle and this is leads to alternating charging and discharging of capacitor.