There is no medium of nutrition, they are continuously replacing the dying cells above then and are being formed continuously as the layer below multiplies.
That is the basic reason that the outer layer keeps dying. And also if these cells were getting nutrition, and some pathogen or mosquito comes and tries to suck out that nutrition, this would have given them a juicy opportunity to infect.
If it was getting nutrition it will never die, then there will be no need to replace. Because of this phenomenon only, hygiene is self maintained by our body by not letting the pathogen take the advantage of our nutrients. This makes the skin the first line of defense!