The amount of Neanderthal admixture shown in Eurasian DNA analysis is similar (1%-4%) for all Eurasians, but there is a definite patttern in which the Neanderthal genes found in Europeans and those found in the Chinese are different. Indeed, it looks like the Neanderthal genes found in South Asians may be different from either of the others.
There are basically two ways this could happen.
One is that we are seeing founder effects from small groups that broke away from a common Eurasian source population that had a certain amount of Neanderthal admixture which was common to the entire Eurasian source population. This would have had to have happened quite early, as the East Asian branch of modern Eurasians breaks away from the West Eurasian branch no earlier than about 45,000 years ago, and possibly as early as 75,000 to 100,000 years ago.
The other is that there was population structure in the proto-European, proto-South Asian and proto-East Asian source populations, each of which had separate admixture events with Neanderthals that were similar in magnitude because the admixture process was similar in each case. This population structure, however, would need to have been present somewhere in the range where Neanderthals were present to admix with these separate founder populations. The admixture of Neanderthals with East Asian founder populations would have to have taken place somewhere in the vicinity of Pakistan, Iran, Turkey, Southwest Asia, and again no earlier than about 45,000 years ago, and possibly as early as 75,000 to 100,000 years ago.
Any admixture, of course, would have had to have taken place by about 30,000 years ago, give or take a few thousand years, when the Neanderthals went extinct.