A quick search found that the populations of China, Japan, and South Korea, are all very homogenous, with nearly 90% of the population being the same ethnic group. Why is this so? Doing further research, I've found that travel between the Japanese Archipelago, Korean Peninsula, and Chinese Mainland occurred often enough that the paths couldn't have been too treacherous for travel. There was limited diplomatic relations between Japan and China by 57 AD. Japan sent imperial envoys to China many times, and had sent large military forces to the Korean Peninsula to fight the Korean kingdom of Silla. Thus, movement between the countries wasn't technologically impossible. So why hasn't there been large scale population movements in East Asia, similar to the Period of Migration in Europe? Why aren't there significant groups of minorities and a lot of ethnic mixing?
Why are East Asian countries ethnically homogenous?
Short answer / TL;DR — They aren't.
Longer answer — I can't speak for Japan, but at least in China, the number that are being reported are not accurate. To start, the People's Republic of China has 56 official ethnic groups, the term often translated as "nationalities" in English. The largest group, the Han (what you'd think of is standard Chinese) are significantly larger than all the rest, but most scholars would agree that you can't trust the numbers as they're reported by the state. What's more, the other 55 ethnic groups are not appropriately divided. For example the Gaoshan ethnicity is actually a grouping of over a dozen distinct Formosan aboriginal groups who live in China. By official counts in China, those are all just Gaoshan. Outside China, in Taiwan for example, each is recognised as their own group and each has their own language, culture and sometimes different ancestral origins. Another similar example in China is the Yi ethnicity, which is actually a few different groups with mutually unintelligible languages and differing cultural traditions. You have many groups like this were a single official ethnicity is actually a group of ethnicities which on paper were bunched together by the Central Government.
Then, within the Han ethnicity, you could actually argue that that is a supergroup, an umbrella term for a collection of related ethnic groups like the Hakka or the Wu or the Cantonese. While genetic studies have shown common ancestry, the same could be said for any other two related ethnic groups anywhere in the world.
Those official numbers are almost certainly fudged by the state as well. If you are Han and you marry someone who's Zhuang and have a kid, you pick one and that's what goes on their ID card. Usually that's going to be Zhuang due to the benefits of China's form of affirmative action, but not always. Also, for ethnic minorities, things like the one child policy aren't as strict, and often not applicable at all. A Hui can have 5 children without anyone batting an eye.
A Han and a person belonging to a minority get together and have a minority baby. Minorities are less restricted as far as population growth. And yet the reported numbers have not shown any major decrease in the population percentage that is Han over the past decades. These factors are often cited as evidence that the numbers we're seeing are not to be trusted.
why hasn't there been large scale population movements in East Asia
There have been. Almost all of the parts of China that speak a Chinese language other than Mandarin are all migrants. Continuously through history, a northern group was displaced by another, and the first moved south, displacing whatever populations existed there before. You have groups like the Dai (Thai) in China who are the remnants of an earlier larger and more indigenous population of Thais, the majority of whom moved south into what it now Thailand. The same goes for the She (畲) or the Hmong (Miao in Chinese).
Why aren't there significant groups of minorities…
Addressed above, but the short answer is that there are, except in places like Korea where it's so small that successive groups have mostly absorbed previous groups. In what's now Japan, you'll see a bit more diversity just because the geography lends itself better to that sort of thing.
…and a lot of ethnic mixing?
This is actually a pretty major point. There is a lot of ethnic mixing, or was, which is why you ave groups like the Han in China that are "pure Han" and Koreans that are "pure Korean". Those are effectively meaningless terms. In modern Taiwan, something like 90% of the population that was on the island before the 1940s has at least some aboriginal blood. There's an incredibly good chance that a Korean person selected at random will have at least some ancestry in groups further north like the Manchus or Mongols or Jurchen or Han or Japanese (zainichis, for example). The Sillabang (Korean communities in the Yangtze Delta region during the Silla dynasty) most certainly left genetic marks and took some of their own.
It's exactly because there's been ethnic mixing that you don't see such distinct groups in places like Korea today. The self reported 90%+ Han of China is not taking into effect mixing.
Adding to what /u/keyilan said, Japanese ethnicity, generally speaking, is mix of Korean/Mainland ethnicity and Ainu population. During history of Korea, people from fallen Korean kingdoms, such as Goguryeo and Baekje, escaped to Japan to avoid execution. Even in, end of Goryeo Dynasty, many Buddhists moved to Japan because Korea forbid Buddhism when it became Neo-Confucius. Even before Korean Kingdoms, Korean ethnic groups migrated into Japan. /u/keyilan is right. East Asian Countries are not ethnically homogenous. It only looks like it because all ethnic groups had been absorbed into one.