First, it looks as if you worked it out correctly. So it's just your niggling worry to address.
Seems to me to be the same underlying idea as Kirchhoff's law.
In circuits, as opposed to static electrics, charge does not accumulate. Charge into a node equals charge out.
Here treating each component as a node.
Charge flows through C1 until the charge gained by one plate and lost by the other is enough to balance the charging voltage. No net charge is gained or lost. You calculate the shift of charge from one plate to the other.
When the switch is changed, charge flows from one plate to the other, through both other capacitors. No charge accumulates on either capacitor. As charge flows into C2, it equally flows out of C2, then into C3. Neither can it stay there, but flows equally out of C3 and back to C1. Nobody gains any net charge. It's the same amount of charge going round the circuit at all places.
If you had resistors in series which charge flowing through driven by a battery, the same charge flows in all parts of the circuit. If 1 Cb flows out of the battery, then 1 Cb flows through each series resistor, no matter how many there are! If you have ten resistors in series, each still has 1 Cb flow through it. 1 Cb out of the battery but a total of 10 Cb through the resistors?
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At the risk of being banned for life from PF, a little hydraulic picture that has nothing whatsoever to do with electricity.
The curved lines are elastic membranes and the values by the boxes show how much liquid has to flow into stretch the membrane enough to raise the pressure difference by 1 Pa. (By magic, my membrane behaves linearly, like a piston and Hooke spring, which would have been harder to draw.)
Image edited
At the start I've pumped 48 ml into the LH device. The other two have unbent membranes.
When the valve is opened, 16 ml flows out of the left device and
both the right devices gain 16 ml.
In losing 16 ml the pressure difference across the first device fell from 12 to 8 Pa. In gaining 16 ml, the pressure across the other two devices rose by 5.33 and 2.66 Pa respectively.