As a Kurd I think majority of Kurds are convinced that the Medes are their ancestors but majority of historian reject that claim, this made me to search a little bit about this topic.
Lately I've found some Syriac Aramic sources that kept mentioning the Medes as a people till Islamic invasions, even in Bible the Medes aren't extinct people as mentioned in Acts 2: 8-11 Then how is it that each of us hears them in our native language? Parthians, Medes and Elamites; residents of Mesopotamia, Judea and Cappadocia, Pontus and Asia
I have more sources that mentioned The Medes, mostly mentioned them in the geography that Kurds live in today, another confusion thing all these Syriac sources above have mentioned Kurds too, so The Medes and the Kurds were living at same time, a Kurdish historian Soran Hamarash believes that The Medes were elite ruling family of the empire, so even after the fall of their empire they kept ruling small kingdoms for centuries later, it's just like today as we mention Ottomans but in fact Ottomans were only elite ruling family and the rest of people were Turks, Kurds, Armenians, Arabs.. etc.
I want to know why they kept mentioning the Medes although their empire fall centuries ago, did the Medes survived or they just referred everyone who lived in Ancient Media as Median?
Sorry for my English, it isn't my first language.
First, there probably never was a "Median Empire", almost certainly not one of the extent suggested by Herodotos. More likely, there was a confederacy of a number of tribes in Northwestern Iran. The idea that Cyrus was subject to them is mainly derived from a Babylonian source that seems to call Cyrus "the Subject/Servant" of Astyages - but the source could just as well be calling Cyrus the servant of Marduk, it's ambiguous. I wouldn't be surprised if Cyrus and the Teispids paid tribute to some tribes to keep them from raiding his borders or something, though. It is certainly conceivable that "Mede" may have been a term used by nobility to self-refer, but between the Achaemenids, Seleucids, and Arsacids, it would really surprise me if that survived into the early middle ages.
I have thought a bit about this over the years, and I think it is mostly a geographic indicator at that point, and quite possibly used similarly to "Kurd" in Middle Persian - simply meaning mountain-dwelling pastoral seminomad, particularly one who speaks an Iranian language. Another example I know of is Attila's supposed claim to be "king of the Huns, the Danes and the Medes". When Tertullian spoeaks of Christian Medes, it wouldn't surprise me if he's talking about, say, Georgian converts.
It's not correct to say that "Kurds descend from the Medes" as the nationalistic claim goes, but it's also not correct to say that they do not. There were a large number of Iranian dialects spoken by people living in what is today Kurdistan according to a pastoral lifestyle; some of these people would certainly have been related to Median tribes. But they would also have intermingled with migrating Turkmen and Arabs - some Iranian tribes adopted Turkish, a lingua franca in some regions, and we know them today as Azeris.
It's worth noting that Christian scholars in Mesopotamia and Iran were pretty well aware of classical antiquity and, ironically, remembered the Achaemenids and so on much better than the Sasanians ever seem to have to. So them making references to people in archaic terms is not in itself that surprising.