Just to put it out there, red hair is not at all a uniquely European trait, and in fact it appeared in Africa first.
we are talking about a much later period in time, no more then about 20,000 yrs ago give or take, long after the 'out of africa' migrations. talking of the red hair trait (resurgence?) that occurred around the same time as the gene for blue and green eyes appeared. all mainly (hypothesised at) developed as a developments in response to the climate/lack of sun that led to lighter colored skin tones
the Tarim Basin mummies -some with red hair and many with accessories in grave that share design traits with other indo aryan (hindu,sycthian,celt,etc) designs vs ancient "chinese" or Turko-Mongol ones, are from around 2000 bce. it is theorized that it was not until this time that the people learned this type of mummification process, probably form neighboring asian groups, but were in all likelihood, living in the area for a much longer time. remains of them, not as well preserved due to diff burial practices.
(these ppl most likely thrived in central asia and are believed to have intermixed with asian Turks and Mongol types who, even to this day, can be seen in the populations of the Uighurs and other Turkic tribes.)
there are also large numbers of ppl in the indian sub continent of all tribes that were 'infused' with dna from the aryan invasions.
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the neanderthal are believed to have had large numbers of ppl with red hair as well, but that is not (so far) connected to the modern red hair gene
if it was, then we could say that everyone with red hair is possessed of neanderthal dna when that may not be the case for all caucasians and asians. modern africans are the only group of ppl (to date) that have been found to have no neanderthal dna-so the red hair is not always the 'same' red hair