My turn to display my warped understanding of electromagnetism:
Putting ANY form of DC into a xfmr would saturate the core of the xfmr.
First - we seem to to be mixing power transmission (DC vs AC) with electronics. DC = direct current and indicates that there are fixed polarities in which the current flows only one direction. Alternating current is named because the current flows one direction and then stops and flows the opposite direction. In the early days Edison was a supporter of DC transmission and Tesla was a supporter of AC. As described in a few posts, the ability for transformer to transfer power to higher voltages (same power with lower currents) allowed AC power to be transported farther and much more efficiently than DC. For this reason Tesla and AC is now used worldwide to move power from the location it is generated to the markets that are using it.
In electronics, AC is mixed with DC in virtually every type of manufactured entertainment device. This is because in the old vacuum tubes as well as modern semiconductors - the current in these devices only operate in one direction (DC). In this case the current in one direction is changing according to the AC signal - that is current in one direction is varying by the amount of the A/C signal. One method used in the electronic circuits to separate the AC signal from the DC is to put both through a transformer. Since only the AC passes through transformer, the signal be removed from the DC.
Too much DC (current) can saturate the transformer (the magnetic field reaches a limit that will not vary with the AC signal component) but it is the exception that is avoided.
To increase DC voltage, use a DC/DC convertor
While the outside of the package has DC in and DC out, what is really happening is DC -> AC -> Transformer -> New AC voltage -> New DC voltage.
Maybe we're talking apples and pears. The source voltage applied to the circuit is a constant. The effective voltage over the capacitor can change. As the charge builds up on the plates, there is a voltage drop in response to the existing charge and the is discharged at an effectively higher voltage.
What is significant though is the capacitor stores electric potential energy and that energy comes from the source (constant V). The benefit of a capacitor and the reason for the flash is the rapid discharge of the capacitor meaning the energy that is stored over an extended period is released quickly meaning a low power is turned into a high power for a brief time.
Capacitors are much like a battery in that they can store a charge. When connected to a DC voltage, current will rush in until the voltage equals the source DC voltage. If the DC varies, the capacitor will discharge or absorb current to match the DC voltage. This property can be used in two ways:
1) If you put one side of the capacitor on a DC that has ad AC signal included and measure the current (typically through a resistor as voltage) the AC signal has been separated from the DC. Capacitors are used routinely in electronics to separate AC (signals) from DC.
2) The ability of capacitors to store energy allow it to slowly accumulate energy from a limited power source and the released it quickly but the voltage and accumulated power do not change. When the capacitor dumps its load suddenly, it is basically short circuited through a transformer. The accumulated energy is transferred to the coil that is wound to provide a high voltage out (in a car - this is the ignition coil) which fires the spark plugs, flash tube, laser or lifts the cow a few inches from the ground (depending on what you are doing with the pulsed high voltage)