With regards to Okinawan karatedo, the link to its Chinese origins are not as obscure or hidden as it is in Japan and Korea. In fact, karate was originally written as 唐手, meaning "China Hand" (pronounced Tode/Toudi in the Okinawan dialect). The Okinawans also referred to their martial arts as kempo 拳法, the Japanese pronunciation of the characters for quanfa (another generic name often used for Chinese martial arts). The name kempo itself was used both interchangeably and together with karate in Okinawa for many years, until karatedo 空手道 became standardized on mainland Japan. Trade between China and Okinawa is well documented, as is the long-standing influence of Chinese martial arts, whether we're talking about Shuri Te (and its descendants) derived from Matsumura Sokon's study of quanfa or Naha Te (and its descendants) derived from Higashionna Kanryo's study of Bai He Quan 白鶴拳. In fact, Goju Ryu, the most prominent descendant of Naha Te, was even awarded honorary koryu status by the Dai Nippon Butoku Kai in 1998 (although the term koryu specifically refers to Japanese martial arts founded before 1868) because of its long, unbroken line of transmission and history.
Moving onto Korean arts, the most common martial arts practiced (Taekwondo and Tang Soo Do) come directly from Japanese karatedo. Tang Soo Do itself is just the Korean pronunciation of 唐手道 and is in essence simply Shotokan (with some Taijiquan influence if we're referring to Moo Duk Kwan specifically). It's only an ingrained hatred for anything Japanese (perhaps rightly so) that forces Koreans to deny this connection and claim Taekwondo is some indigenous art supported by 2,000-year-old cave drawings, when in fact Taekwondo was created solely because Tang Soo Do was "too Japanese." Even the fighting arts derived from the MooYeDoBoTongJi, a Korean military manual from the 1400s, can be traced to China and the Shaolin temple. The manual itself is pretty much a copy of an older Ming Dynasty military manual, Jixiao Xinshu, written by General Qi Jiguang. As if that wasn't conclusive enough, the MooYeDoBoTongJi contains an entire section of kwon bup, which is simply the Korean pronunciation of quanfa.
In Japan, the waters begin to get a little muddier, but why is that?
Getting back to the Chinese connection (no, not the Bruce Lee movie), the Chinese and Japanese first made contact around 200AD according to Chinese records and whether you look at the language, culture, political institutions or religion, the influence is apparent. The close of the Sengoku Jidai (Warring States Period, 1467-1603) and start of the Tokugawa shogunate (Edo Period, 1603-1868) saw a resurgence in Chinese contact and influence. Ironically, right around the same time many of the koryu martial arts began to become systemized into the ryu we know today.
From a technical perspective, jujutsu and qinna 擒拿 are nearly identical. While there are stylistic differences and preferences, the underlying principles make these two arts more like siblings than distant cousins. Both arts revolve around joint manipulation/destruction and throwing techniques, while also teaching a wide array of strikes and pressure point techniques. While it is possible the two evolved independently of one another based upon the universal principles of biomechanics, given the Chinese influence of literally everything else in Japan I find that highly unlikely. Even "purely" Japanese arts such as Aikido have a suspect history. While it's common knowledge that Ueshiba Morihei studied Daito Ryu Aiki Jujutsu under Takeda Sokaku (which became the primary base for Aikido), his travels to Manchuria and reported study of Baguazhang and Xingyi make it hard to identify Daito Ryu as the sole source of Ueshiba's internal power. And then of course Korean hapkido also comes from Daito Ryu so that connection is just a continuation of the same transmission.
Will we ever make a definitive connection between Chinese martial arts and their Japanese counterparts beyond mere speculation and logical progression? Probably not. However, China's dominance over the entire region of Southeast Asia can't be ignored. They led the way in trade, commerce and culture. They were the society that everyone tried to emulate. With nearly every other facet of Chinese culture being imported to Japan, it's only natural to conclude that its fighting methods came too.
We can argue about stylistic differences all day long. There are only so many ways to strike, kick, lock or throw another human being. At their core, all styles contain the same physical techniques and therefore styles are only defined by the principles, philosophies and training methods. At the end of the day, they all share the same essence, the same lifeline, the same history. At the end of the day, it's all gongfu.