How to draw constant current independent of voltage for capacitor testing
You need a constant current source/sink. The one-transistor and zener which are everywhere should be fine (or if you don't have a zener, just a resistor ladder). You can find an example of that on the wikipedia page.

https://en.wikipedia.org/wiki/Current_source

It amounts to a fixed voltage presented to a base of an NPN transistor, a resistor between the emitter and groung, and a source between the collector and supply rail.

The divider at the base presents a fixed voltage above ground to the base which then travels through the BE junction (where 0.6V is incidentally dropped) and this fixed voltage then travels through the emitter resistor to ground. As the voltage at the emitter is fixed by this base voltage, so is the current through the emitter resistor. Almost all of this (hFE is large) current comes from the collector, so through the collector comes a fixed current.

Looking at it another way, (less useful for component values, but more intuitive), should the resistance of the collector increase for some reason, then less current will momentarily travel through the emitter resistor, the voltage at the emitter will drop, increasing the base-emitter voltage turning the transistor more on. And vice versa.

Using this to charge and discharge capacitors is a good use for such a circuit. If you are interested in getting charge and discharge times very long, for example, using a constant-current source/sink allows you to maximise the time for a given RC combination, for a given leakage current, because you are charging linearly rather than "wastefully" quickly at the start.