My wife is just getting into pottery again, and last time she always had someone else's kiln available, so I really don't know much about them yet. But I know the newer controls are electronic, but the older ones had more of an electro-mechanical control. The control is called a kiln-sitter, and that's usually the most prominent name on the whole kiln, so if you see someone selling a "kiln-sitter" brand kiln, it means the seller doesn't know kilns, because that's only the brand of the control.
I believe the kiln sitter has a timer switch built in, three outputs for the three heating elements. The kiln sections are stacked, the bottom one with the kiln sitter control on it, the middle and upper sections have heating elements only. The temperature is ramped up slowly by turning on first one element, then two, then all three. The kiln sitter also has a little switch extending into the kiln itself, that you set a cone of wax on, and when the wax melts, the switch is opened and the kiln turns off. You use different cones of wax for different maximum temperatures. That's the extent of my understanding of it.
I didn't finish wiring the circuit yet, so we haven't tested her kiln yet. I will try to find any amp ratings listed on it.