You actually picked two very different ways of getting eunuchs!

The Ottomans got their palace eunuchs through relatively straight forward human trafficking; they used slaves extensively and in almost every function you can imagine, eunuchs but one part. They had two sources of eunuchs: white eunuchs, who after about 1500 would guard the "male harem," or the entrance to the sultan's family quarters, usually came from Circassia, Georgia, or the Caucus, trafficked through the Mediterranean. Black eunuchs, who would guard the women's harem (inside the larger quarters) would usually come from Abyssinia, trafficked through Egypt. Castration was illegal under Islamic law, so they were castrated before entering Ottoman territory. This rule was not often violated. The black eunuchs would often be bought by other officials, trained up in Ottoman rigmarole, and presented to the sultan as a present, but eunuchs were also bought direct. After the mid 1700s the white slave trade got cut off, so they switched entirely to black eunuchs until the bitter end of Islamic use of eunuchs, in the 1920s or so. On the procedure.

The Chinese got their eunuchs in a few ways. Castration was one of the "Five Punishments," but other than the early days I don't think criminals were very common, who wants criminals? You could enter palace service as an adult or a child castrate. There was a mild preference for child castrates, but ample evidence that it was something adult men chose. There were professional castrators, although it was sometimes done at home, Sun Yaoting, the last Chinese eunuch, was castrated at home. It was usually done out of poverty and desperation, so it was voluntary for the adults, and I suppose voluntary for the children's parents as well. Very different from the Ottoman racial-based slavery at the least. And on that procedure.