I can discuss the multiregional theory for hominin dispersal and the evidence for this theory.
Energy
Joule
Transformation
This theory argues that Homo erectus left Africa around 1.8 mya, and evolved into Homo sapiens simultaneously in Africa, Europe, and Asia.
Regional subspecies such as the Neanderthals developed, but they were all still Homo sapiens.
Gene flow (interbreeding) between each regional populations ensured that they did not develop into distinct species.
Fossil evidence that Homo ergaster was in Europe.
Certain fossils represent intermediates between Homo erectus, Neanderthals and modern Homo sapiens (called transitional fossils).
Intermediate fossils between Homo erectus and Homo sapiens were found in Israel (meaning some gene flow occurred/ close to Africa).
That many fossil finds cannot easily be classified as belonging to one group or the other because they are the same species.
The physical differences between these groups represent regional differences. Eg: Neanderthal were cold adapted.
Interbreeding has taken place in the past because modern Europeans have about 2.5% Neanderthal DNA.
Belief that 170,000 years is too short to develop the racial differences present today.
Mistrust of mtDNA evidence.
High levels of gene flow would be needed for this hypothesis to be correct, which is not very likely.
Highly questionable based on the more recent genetic research, especially the use of mtDNA and Y chromosome DNA.
For this hypothesis to be correct, current humans would have to have large genetic variation. However, when mtDNA (mitochondrial DNA) and Y chromosome DNA is analysed, there was less variation than expected. This means they didn't leave Africa until much later.
I can discuss the Out of Africa Model for hominin dispersal and the evidence for this theory.
Energy
Joule
Transformation
This theory argues that Homo erectus left Africa twice.
At around 1.8 million years ago, they left for the first time as Homo erectus, and go to Africa, Asia, and Europe. Some Homo erectus in Asia evolved into Denisovans and Homo floresiensis. Some Homo erectus in Europe evolved into Neanderthals. Some Homo erectus in Africa evolved into Homo sapiens.
Somewhere around 120,000-50,000 years ago, those Homo erectus that evolved into Homo sapiens left Africa for the "second time" and outcompeted and replaced other hominins in Africa, Asia, and Europe.
Every person who is not of African descent is related to a single group of about 200 people who left Africa and crossed the Red Sea.
These movements were made possible by an ice age which caused sea levels to drop.
When both mtDNA and Y chromosome DNA was analysed, it was observed that the DNA had been highly conserved (hasn't changed much). So it has been predicted that 'Mitochondrial Eve' left Africa much later, about 100,000 years ago.
Genetic similarities across races - All modern humans have similar genes and nuclear sequences. If parallel evolution (Multiregional hypothesis) had occurred, there would be much greater variation in human genetic makeup.
People that live in Africa have the greatest genetic variation as they have been around for the longest time and have had more time to develop genetic diversity. The greatest variability is found within African populations which are the oldest. There is less genetic diversity in Asian and European populations, which suggests they are not as old as African populations.
Also, the Multiregional theory would require gene flow across continents during the last million years, which would be unlikely due to vast distances.