(You can catch Part 1 of this dual post on cosmetic packaging and irradiation here by the way.)
As I would find out while reading up on irradiation, Gamma Rays are produced by a radioactive isotope called Cobalt 60. It produces high energy photons that are disruptive to the DNA and structure of living things. This is why they are capable of sterilizing. The DNA of the microbe is broken down during treatment and it is unable to recover. The FDA limits the energy levels the radioactive material can emit. The permitted radioactivity level makes the gamma rays pass straight through the treated material and leave it with no residual radioactivity. Essentially it isn't strong enough to knock apart the atoms in the treated material and destabilize them. It's sort of like an X-ray that way.
I could write it off if one source told me that, but two unrelated companies were surprised when we asked about appropriate cleaning procedures for closures. According to both of those sources the industry standard is just to use the pumps and bottles straight out of the cases. I was shocked. They may come out of the cases without gunk stuck to them, but the moment someone sneezes or touches them they've got microbes on them, some that may be disease causing pathogens.