But black people didn't stay the same! They adapted too to their environment, through the process that huogestr describes. Look for example at the pygmies in the forests of Equatorial Africa: they are extremely short because they had to adapt to life in a dense forest. The Maasai by contrast are very tall and skinny being adapted to life on the savannah.
Two questions we have to establish before going forward: what are humans and what is black? It may seem obvious but it's not.
As far as humans are concerned, the consensus is that the first hominids appeared in Africa's great Rift Valley around 6-3.5 million years ago. The first hominid to leave Africa was Homo Erectus about 900,000 years ago. Here's were it gets more complicated: Homo Erectus started to evolve. Thus we have Homo Neanderthalensis in Europe and the Middle East who was well adapted to cold climates, and Homo Florensis on the island of Flores. Most researchers believe that modern humans (Homo Sapiens) appeared in Africa again (I heard South Africa), about 200,000 years ago, and spread throughout the world, displacing the other species. Some researchers however believe that from Homo Erectus, other species of humans have evolved, which mixed with the African Homo Sapiens, to give rise to certain populations. I believe that in particular the Chinese reseachers back this theory. So, did hominids originate in Africa? Most definitely. Did (all) humans originate in Africa? Probably, but not certainly.
Now, as far as the meaning of black, supposing that the Khoisan are indeed the earliest humans and that South Africa is the land where they first appeared (it's difficult to say based on their current geographical location, because they used to occupy a much larger landmass in the past), then are the Khoisan one and the same with black people? No. They are African, but they differ as much from a black person from Senegal or Nigeria or Somalia, as they differ from a Chinese. From the standpoint of white ignorant people (or black or chinese ignorant people for that matter), they are black simply because their skin is somewhat dark and they are African. But then again, they are about as dark as say a Phillipino, so is that a fair statement? South Africa is as the same latitude as a place like Buenos Aires in Argentina or Sydney in Australia; or in the Northern Hemisphere, as the same latitude as Texas, Palestine, Iran or Central China, so the whole idea of being "black" for the purpose of adapting to sun radiation doesn't hold much ground. Also, evolution and adaptation means a whole lot more than skin color and appearance. We have a lot of genes, regulating our metabolism, makeup of organs, skeletal system, immune system, etc. The differences between individuals of the same population are often greater than the differences with members of another population. For example, two individuals from Nigeria may be more different genetically from each other than one of these individuals and another individual from Sweden or China. Race is a myth based on a very shallow understanding of superficial differences, which ignores the rest of the similarities and differences. Appearance is like the tip of an iceberg, when it comes to genes.
To summarize, humans probably appeared in one spot and then evolved and diversified through a process of natural selection. Black Africans along with everyone else went through this process. Even the Khoisan probably evolved to some extent, though they may be closest in appearance to our ancestors having stayed put in the original homeland. No one people can claim being the original people, because it appears that we are all descended from a common population. There is no race, other than a very inacurate but quick convention of describing people.