Post = after, tussive = cough, emesis = vomit
Kids tend to have a very active gag reflex, so they sometimes gag themselves and vomit with cough. This can be good, since it gets the mucus out of the back of the throat. You can try to teach older kids to hack and spit it out, cough and spit it out, gargle with salt water, and rinse mucus out of the nose.
Of course it's not fun to vomit after coughing because everything in the stomach comes up and makes a huge mess. Sometimes the vomit comes out of the nose, which can burn from the stomach acid. And vomiting can be very scary to kids.
Are there serious concerns when kids vomit from coughing?
Yes. In medical school I learned that when kids cough to the point of vomiting we should consider whooping cough, pneumonia and asthma. In reality, I find that many kids with regular cough and colds can gag from cough, but I always consider the more serious options.