All evidence thus far that I'm aware of points to the former, not the latter. Pre-modern homo sapiens developed, according to genetic analysis and confirmed thus far by all fossil evidence (though we found the fossils first, obviously, by the Leakey's and others in the early and mid part of last century) in the Great Rift Valley, particularly in the portion of the Great Rift Valley that is in Kenya.@Hodj Is there any consensus if we all started out black and then lost that pigment as we migrated to Europe and Asia or White and the skin darkened over time to protect against the sun?
Dark skin color in bright, hot environments is strongly selected for because lighter skinned people in those environments are more prone to skin cancers, heat exhaustion, and other heat related conditions that can be fatal.
Lighter skin is selected for strongly in colder environments that see less overall sunlight during the year because lighter skin allows for more efficient production of Vitamin D, which is primarily synthesized in humans via physiological reactions to sunlight (I don't know the pathway by which that operates, but you can google like Vitamin D pathway in humans or something and probably get a full explanation of how that works).
This is why you see a gradation of skin tones as you move away from the equator, as our bodies balanced the need for protection from cancer causing sunlight and our need for sunlight to produce Vitamin D.
Anthropologists from the 1800s up until the Leakey and other findings in Africa, looked high and low for evidence that humanity first evolved in Europe, even going so far as to hypothesize we evolved in Britain and the like, but not much evidence has ever been provided to justify that claim.
Now, some of this paradigm might be shifting a bit towards a position that after pre modern homo sapiens spread out around the globe, that modern homo sapiens developed sort of concurrently across all three major continents, which is plausible, but the verdict is still out on that one.