Mineral Oil vs vegetable oils
Jay.O unfortunately it doesn't matter whether its called tanning, cooking or baby - its still Oil and will therefore fry anything in it when heat is applied.
The problem with baby oil is even a little worse than that since it is a hydrocarbon by-product. ie its made from Oil, the black stuff you stick in your car. It is like putting plastic wrap on your skin (or in this case your horse's skin) which tends to magnify the suns burning UV rays and block the skins ability to breath at the same time. BTW, This also applies to petroleum jelly (vaseline)
I have also read the baby oil, unlike most other products has molecules small enough to pass through the outer layers of the skin. This means that it can transport bacteria and\or dirt INTO the skin but I haven't verified this aspect.
Mineral oils have received a lot of press recently because it is very bad for babies, small humans and anyone\thing that uses large quantities of the stuff because once the oil is absorbed, it is broken down by the liver and passes through the intestinal tract. And once the oil is present in the intestinal tract, it will absorb all of the fat-soluble vitamins found there. It is essentially stealing important vitamins from the body, which the body will not be able to replace. This can eventually lead to nutritional deficiencies.
Veg oils on the other hand are easily absorbed and contain vitamins of their own so don't have the same downside.