The following is taken directly from Dale Ripke's website;
Lemurians:---the race of Lemuria. After the destruction of their land, the Lemurians fled to the eastern shore of the world-continent where the ancestors of the Khitans enslaved them for many centuries. Later they were able to revolt successfully against their masters and migrate westward. This migration may have been prompted by racial memories of the ancient colony of Lemuria-of-the-West. There were in ancient times two main lobes of Lemurian movement: the southern, which eventually founded Old Stygia, and the northern, which founded Acheron. A much later movement of Lemurian peoples out of the deserts and steppes of the Far East gave rise to the Hyrkanians. (Hyborian Age I, Mirrors of Tuzun Thune)
The Khari, the non-Thurian rulers of ancient Khitai, were the founders of modern Stygia -- and presumably Acheron as well, though Acheron may have been co-founded with the pre-human "Giant Kings" that ruled Stygia before the arrival of the Khari). Common mistake to assume they were Lemurians...
The descendents of the Lemurians, mixed with remnants of the Khari, gave rise to the modern Khitain, while purer Lemurians (disdaining the remnants of Khari civilization) migrated west, where they became known as the Hyrkanians. There they presumably mixed with the remnants of the peoples of Grondar, the easternmost civilization of the seven Kingdoms, destroyed in the Cataclysm.
The Hyrkanians eventually moved further west, north and south of the Vilayet, and founded Turan, where they mixed with native Shemite, Iranistani, far-wandering Hyborians (such as Kothites), and undifferentiated local tribes to form the modern Turanian race. Hyrkanians also invaded ancient Vendhya, and formed the core of the Ksatriya (sp?), or ruling caste.
As for any remnants of the Lemurians in the isles that remained in the Eastern Ocean... that is up to you, as Howard never really delved into them. Some presume they were completely destroyed, others have made up their own various races and cultures. I'd say "Japanese" types would be migrants from Khitain, while local remaining Lemurians would ahve evolved into something like Polynesians...
Any lemurian desendant's remaining on whats left of leumuria would, if there still Human, more likely resemble the Ainu than the Polynesians.
Also the civilisation, if there is one, ruleing the lemurian isles might have come from Mayapan like the Khari did instead of Khitai.