Regarding transmission in China I had a topic somewhat relevant to the question: How Important Is Transmission?
There are numerous studies (well, not too much) on Chan history covering the time from Bodhidharma to the 20th century, although I haven't yet met (which doesn't mean there isn't) anything substantial on the Yuan era (1271-1368, Mongolian rule) or the Qing (1644-1911, Manchu rule).
Even though historical records show that the internal (religious) history of Chan is far from being accurate, it doesn't mean there were no Chan followers - ie. at some point it stopped being practised - since its rise around the 11th century. Chan became an integral part of Chinese Buddhism, its end would mean the cease of Buddhism. There were of course periods when Chan wasn't popular, nevertheless, popularity is not a requirement for existence.
As for the survival of lineages, just as Dexing says, both Linji and Caodong exist in China. The Caodong tradition actually has a stronger claim for historical continuity as it was preserved in the world famous Shaolin monastery and spread out from there in the Ming dynasty Chan revival. Although that doesn't mean it is different from what is practised in the Linji lineage. Also, another source of confusion is that people mainly here about Japanese Zen, which is a separate development with its own 800 years of history. Another noteworthy phenomenon is that Ven. Xuyun (1840-1959) - the most well known Chan master of the last century - is said to be a lineage holder of all five schools of Chan, that's how for instance Ven. Xuanhua (founder of the City of Ten Thousand Buddhas) was in the lineage of the Weiyang school (which didn't survive beyond the Song dynasty, so there is a leap/gap in the transmission).
An important thing to understand about East-Asian Buddhism is that there is very little or no sectarianism at all, unlike in Japan (which has its historical reasons of course). Thus you find Chinese teachers passing on many forms of teaching: Chan, Pure Land, Esoteric, Vinaya, Tiantai, Huayan, etc., while at the same time they may be specialist in any of them. If I remember correctly, Hashang Mohoyen is credited not only with teaching Chan in Tibet but also many sutras.