There's no acai in Saint Lucia to put down locally? (6)

I believe the answer is:

insult

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'down locally?' is the definition.
The answer and definition can be both acts as well as being singular nouns.
Maybe you can see an association between them that I don't see?

I don't understand the remainder of the clue.

Can you help me to learn more?

(Other definitions for insult that I've seen before include "A deliberately offensive remark" , "Treat with contempt" , "Slight - affront" , "Abuse (someone)" , "Disrespectful remark" .)

I've seen this clue in the Irish Times (Crosaire) and The Irish Times.

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