When we look at the etymology of a word, we are essentially asking where each part comes from. Specifically, the origin of a word and how its use has changed through history. In English, word origins can be from a lot of countries.
For example, the etymology for destroy is originally from the Classical Latin word dēstruō, from dē- (“un-, de-”) + struō (“I build”). It evolved into a slightly different spelling in Vulgar Latin (vulgar meaning common in this case), then another one in Old French, then to something else in Middle English. From there, it was picked up in modern English.
Etymologies don't necessarily have to involve that much change. The etymology of the verb Shanghai is from the Chinese port Shanghai. Apparently, in the 1800s, shipping owners on the US West Coast would force people into the crews of ships bound for the Far East, typically for Shanghai. Turning nouns into verbs, by the way, is longstanding tradition in English. I think that Shanghai isn't common in modern English (the specific practice is extremely illegal and also a terrible idea even apart from the ethics and legality). As an aside, a lot of English words of Chinese origin may follow the Wade-Giles romanization (an older one, whereas modern speakers prefer Pinyin), or they may not be from Mandarin (e.g. we say wok in English, which is of Cantonese origin minus the tones).
There's no single relevant verb, I think. You would just ask, "what's the etymology of (whatever word)?"
The other answer is very interesting, and I think it is more technically correct. However, be aware that this is a technical term. Linguists would understand it. However, biologists would understand it a slightly different way. And lay people might guess what you mean from the context, but we also might not.