This conflicts with some advice I have previously heard, so would be interested in any hard research for or against.

I was told that if you vomit, you should not brush or rinse your mouth. You are only moving the low pH stomach acid around your mouth and exposing more teeth to the acidic environment. Instead you should try to neutralize the acid (baking powder or similar).

On one hand, I could believe you are just spreading the damage around. On the other, it feels hard to believe that some residual ~2 pH stomach acid would remain when massively diluted by a mouthful of ~7 pH tap water.