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Unless a breakdown voltage (voltage at which current will flow across the dielectric) is reached, charge doe not flow across the plates of a capacitor. If the plates are shorted (connected), charge flows out of the plate. I'm assuming the capacitor is ideal.
There is a dielectric or not, unless there is no dielectric breakdown no charge can ever flow from inside a capacitor. What we mean by current flowing through a capacitor is that + charge is getting accumulated at the positive plate and negative charge is getting accumulated at the negative plate. In other words, electron are leaving from the positive plate and electrons are entering the negative plate and hence we say that a current is flowing "through" the capacitor but in fact there is current from within the capacitor itself.
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Actually, charge can flow across the dielectric without exceeding the breakdown voltage. It's called leakage current. If you charge a capacitor then leave it alone, it will eventually discharge itself. Ideally though, current doesn't flow through a capacitor. Basically, electrons collect on one side of a capacitor which displaces the charge on the other side of the capacitor through the force of its electric field. This apparent charge flow is called displacement current, although no actual current flows through the dielectric.
I hope you mean how the capacitors are charged up when dielectrics can't conduct the charge? Let's start with a bit more easy example. Assume you have an air capacitor with one plate connectet to the ground. What happens if you charge the second plate with any charge (let it be a positive one e.g.)? If it is positive now, it can affect another charges. Are there any charges that can be easily affected? Yes, the electrons in the ground. They are attracked by the positive charge and flow until we have a charge of the same magnitude but opposite sign on another plate of the capacitor.
The same is for two capacitors in series. One of the open ends (plate 4) is still connected to the ground, and another one is positively charged. By induction, this charge attracts electrons from the 3rd plate, and now we have a lack of electrons there, which makes plate number 3 a positive one. Electrons from the ground flow to the 4th plate as in my first example.
The same chain reaction can be applied for a serial connection of more than 2 capatiors.
Hope it helps!