In a switching topology, and I am butchering it now, the inductor gets "charged", and eventually will keep an amount of energy "just enough" required for the load. This is achieved by the control circuit and the switch (in any general topology) stops to charge the inductor when the output voltage, filtered by the output capacitor, goes higher than a certain predefined value. Then this excess of energy gets transferred to the load, and the cycle begins. In this way, the energy transfer is chopped over time to fullfill the needs of the load, instead of being dissipated away.
When the inductor charges and discharges, this results in linear current variation flowing through the inductor itself, in which the rising ramp current (over time) translates in the energy took from the input rails, while the falling ramp of the inductor current (over time) corresponds to the energy given to the load.
The output capacitor present on most common topologies, is taking this energy to be provided to the load and forms an LC filter to achieve constant voltage.
So in this way, there is this game of energy taking and giving, by exploiting the L and C foundamental properties (given by their characteristic equations). This two components, L and C, are reciprocal, what works with current in one, the same happens in voltage on the other.
So my question is: Can a switching topology, let's say a buck converter, with the given adjustments, use as input storage element a capacitor instead of the inductor, and an inductor instead of a capacitor? And why are not used?