They can get radioactivated, like you describe – either an alpha capture or a neutron capture can turn a stable nucleus into a radioactive nucleus.
Or, if the two materials actually touch, breakdown products can transfer from one to the other and so could the original material, if it’s soft enough or if it’s dissolved/powdered.
I was reading about this years ago after watching the Adrian Edmondson movie, where he describes the graphite moderator as “just…unbelievably radioactive”. The neutron flux in the reactor does convert some of the graphite into carbon-13 and carbon-14, but I think the bigger problem was fission products like strontium and barium sticking to, or soaking into, the graphite…? Carbon-14 just doesn’t seem that scary to me because of its long half-life.
[Edit: Iodine, too.]
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