Switched capacitor resistor
You are right, if we assume that the input is driven by an ideal voltage source and that we have ideal switches then the circuit can not be analyzed using our standard definitions of circuit elements. Connecting two voltage sources with different voltages in parallel results in an invalid circuit because it violates our definition of what "parallel" means.

At the instant after the switch closes the voltage on the capacitor must be exactly what it was at the instant before the switch closed. An instantaneous change in voltage across a capacitor would require an infinite current, which once again makes the circuit invalid and impossible to analyze.

Power is most certainly consumed in such a circuit, but it is consumed in the resistances of the real-world switches and real-world voltage sources rather than in the capacitors.