The description suggests the protocol -- if I were doing it, I'd be interested in the short-term (self-healing) noise voltage, and the average leakage current, with respect to time, voltage, temperature, thermal history and thermal cycling. (That's a lot of variables, yes; it would be very involved to test all of them exhaustively, all the while taking a statistical sample of components! Just a few combinations would be fine.)
The noise measurement would look very much like a Geiger counter: the expected mechanism is impulsive, so amplifying the AC voltage, and logging the magnitude of the impulse, and rate (presumably following Gaussian and Poissonian statistics, respectively?), and how they vary with various conditions, would be interesting.
And, actually logging that data as such (say, magnitude and time stamp), would be a good random number generator, but not very useful given the amount of effort required to collect it (essentially a photomultiplier spectrometer -- an impulse size and rate counter, but slower), so simpler methods could be used, like taking the AC RMS voltage over a suitable duration (it might not be something you can measure with a TRMS DMM, but an RMS converter chip and a minutes-long filter would do).
It would then be interesting to correlate noise with DC leakage, and DC leakage with time, and DC leakage or integrated noise against capacitance change (if any). Capacitance change is quite pronounced in self-healing film caps, but probably not that strong here? Would be an interesting hypothesis to test.
Standard testing -- and expectation -- for capacitors in general, is applying a voltage via current-limiting resistor, and monitoring the leakage current over time. There will always be a long time constant due to dielectric absorption (which may not be a time constant at all, but a 1/sqrt(t) -- diffusion -- dependency), but there may be additional effects at work. They can be detected by teasing apart the different responses stacked on top of each other, assuming simple RC time constants, and/or a diffusion element.
Presumably, there will be some reforming or self-healing effect, i.e., the leakage starts high then falls gradually, at a given voltage; subsequently, that voltage (or below) will retain a low leakage level, but raising the voltage higher causes more leakage, until that level is "formed", and so on.
The test can be accelerated somewhat, by using a low impedance driver. The current should still be limited to low levels, but the dynamic impedance (change in voltage / change in current, for small changes) can be quite low. A good example is an op-amp follower, set for a constant input (and therefore output) voltage, and measuring its output current. That way, as soon as leakage is less than the current limit, output voltage will be forced to exactly the setpoint, eliminating the potentially long or compounded time constant an RC test has. Downside: output noise voltage is
differentiated into noise current (Inoise = C * dVnoise/dt), making this a difficult method to combine with an AC noise test (depending on how much excess noise the DUT has, relative to the amp).
But yeah, the basic test would be logging leakage current (hopefully microamperes) into a plot, for a bunch of individual parts, and raising and lowering the voltage stepwise to see the various effects. Then the temperature stepwise, and so on. Take as many steps as you like, each one as long as needed to verify the time constant(s).
Tim