This rash occurs usually on the trunk and perhaps the nape of the neck. Those little sweat glands work fiendishly, pumping out sweat - which has nowhere to go, since the pores are stopped up tight with old skin cells, dried oil, and so forth. So the rash is fine and bumpy - the miliaria part of the name.
Treatment is dead simple.
Next, when you bathe little Buddy, use a coarse washcloth to rub the skin a bit and open up the pores. I can't remember how well that worked, but Moms knew lots of good stuff back then. Once the pores are open, no more opportunity for the rash. That is why we don't see it in the summer, when light clothing and profuse sweating are de rigeur.