Cheap, compact 3.7VDC -> 400kVDC (continuous) converters
I do not think there is DC to AC conversion. Most probably they are just stepping up the voltage step by step like in the classic Cockroft-Walton circuit. A clock is needed for that, but i would not consider it as AC conversion. A step-up or boost converter could be used as well, where the idea is that if we charge up an inductor and switch off its output load (= generating an open-circuit), than its output voltage could be in theory infinitely high, since the current of the inductor can not flow anywhere. In reality there are always leakage paths and a voltage limit where the some isolating material (even air) breaks through. Such a configuration is seen in this tear-down video.

There was a small series about charge pumps in the IEEE's SSC Magazine and also an overview in Circuit and Systems Magaizin. They are really good reads. Check it out if you have access to it. (Universities and libraries usually have)

T. Tanzawa, “Innovation of Switched-Capacitor Voltage Multiplier: Part 1: A Brief History,” IEEE Solid-State Circuits Magazine, vol. 8, no. 1, pp. 51–59, winter 2016, doi: 10.1109/MSSC.2015.2495678.

G. Palumbo and D. Pappalardo, “Charge pump circuits: An overview on design strategies and topologies,” IEEE Circuits Syst. Mag., vol. 10, no. 1, pp. 31–45, 2010, doi: 10.1109/MCAS.2009.935695.

About the step up or boost converter, just google it. There are many explanations.