Yes it is safe to blast the parts in a cabinet because you will be able to thoroughly clean them before reinstalling.
The question is..... What do mean by "media"? Media usually refers to the plastic blasting material used at high pressure typically used to strip assembled cars of paint only.
"Media" is not aggressive much at all. It will not remove rust and corrosion etc.
You should probably consider glass bead or sand or grit of some sort to clean up the parts mentioned. Glass being the least aggressive of those mentioned.
Good point. I was referring to "media" as the various sorts of material people use in blast cabinets.
The good news is there are a lot of options for media now when trying to clean parts.