Do you say Kid gloves or kit gloves?

Both terms are actually correct. Many purists will argue the fact that the original term "Kit Gloves" is correct, but more people recognize "Kid Gloves" as the proper term. "Kid Gloves" according to Websters is defined as a term, and has come to mean treating something gently, as you would wearing soft goat skin gloves.

The original term, "Kit Gloves", meant to use the right tool for the job, or doing something correctly, using the right "kit" to do the job.

To the "purists" who argue vehemently that the original term "Kit Gloves" is correct, I will only say this: Have you ever used the term "Selfie" to describe taking a picture of yourself?

The Word "Selfie" was officially recognized by Websters a few years ago, making it an acceptable, defined Word. Since "Kid Gloves" are now recognized as a term in Websters, and most people polled agree, it makes the argument irrelevant. If you want to get really specific, use "Kid Gloves" for doing something gently, and use "Kit Gloves" if you want to mean to do something correctly, or using the right tool.