I am using a KL divergence to measure the difference between distributions, but I would like to compare the results to each one another, because the probability distributions I'm measuring are correlated. What is a good measure for comparing two KL divergences on related distributions?

Ex. If I'm looking at some characterization data for a test, let's say X-ray diffraction patterns for a metal sample. I get an XRD pattern of intensity vs. diffraction angle, and can create a probability distribution for that in a reference state (room temp, new sample). I then run the same XRD test on the same sample while I vary temperature, and for that temperature I get a new distribution of my XRD data, and I can calculate the KL divergence for that distribution relative to my reference one.

Let's say I then cold-roll my metal sample, and run the same XRD experiment, and then calculate the KL divergence with the same reference state for those distributions. I have two KL divergence results, and I know that there are variables for the metal sample, temperature vs. strength, that are correlated. Is there any comparison technique between divergences to tell me how much one distribution diverged relative to another? Especially if the variables are interrelated?

Edit: Changing the example to be more specific.