Later on when Cyprus fall under several other rulers, the populations that those rulers brought to Cyprus was also gradually assimilated into (Greek) Cypriots.
So (Greek) Cypriots are as indigenous to this island as it can get. They didn't invade, kill and steal this land from others, like say Turks did with Asia Minor and are now trying to do with north Cyprus.
The Turks have occupied Cyprus for less time than they occupied Athens and most other Greek territories and islands. The Turkish minority that was created here is not unique either, since such minorities where created in many other territories the Ottomans had occupied, and not only Greek ones. (E.g. Bulgaria has a 12% Turkish minority).
Of course Cyprus and Cypriots have their own unique local characteristics, in the same way that people from north Greece have their unique characteristics different from those in Athens or Pelloponisous, or Cretans have their own unique characteristics (more similar to Cypriot, than say Athens) etc. This is the same in France, England etc, different localities can have different unique characteristics.
. Australians speak English and most have their roots in England also Americans speak English and lots have roots in England but they have grown up and left mothers nest and do not now hang on to her apron strings. They are proud to be Australian and American and rightly so.
What you forget though is that those people choose to have their own separate countries. Otherwise how can you explain say, Falklands? Why Falklands are part of the UK? Or even say Manchester (with a population way more than Cyprus), why should that be part of England?
The point here is that it should have been up to Cypriots to make their own democratic choices about their own island, not up to some foreigners to impose on us their will.
After Greece had won its independence from the Ottoman Empire in 1821, the idea of enosis (union with Greece) took hold among ethnic Greeks living in the Ionian and Aegean islands, Crete, Cyprus, and areas of Anatolia. Britain ceded the Ionian Islands to Greece in 1864, and after control of Cyprus passed from the Ottoman Empire to the British Empire in 1878, Greek Cypriots saw the ceding of the Ionian islands as a precedent for enosis for themselves. Under British rule, agitation for enosis varied with time. After World War II, in the era of the breakup of colonial empires, the movement gained strength, and Greek Cypriots spurned British liberalization efforts. In the mid-1950s, when anticolonial guerrilla activities began, Turkish Cypriots--who until that time had only rarely expressed opposition to enosis--began to agitate for taksim, or partition, and Greece and Turkey began actively to support their respective ethnic groups on the island.
This is very interesting actually because it supports what I have been always saying and I have shown in this thread:
http://www.cyprus-forum.com/viewtopic.php?t=14652
The TCs themselves up until then knew that union with Greece was the natural result of Cyprus being decolonized since the vast majority of the population of Cyprus was Greek and union with Greece was what they wanted and what was their right. The idea of partition and fighting against Greek Cypriots was part of the divide and rule game that the British played against Cyprus. They used the TC minority as a way to deny to the Cypriot people their freedom and self-determination. Watch the second video from the link above where a British diplomat goes to Turkey to propose to them the partition of Cyprus.
About Turkish Cypriots, and the how native they are, that depends on who they really are. According to some sources TCs are those Ottoman soldiers that invaded Cyprus and later brought in their families. According to other sources, most TCs are Cypriots who changed religion and became Muslims in order to enjoy the benefits of such a change.
If the first hypothesis is true, then TCs are not native to Cyprus. After 400 years living here they are not foreigners of course, but still not native. If the second hypothesis is true, then TCs are as native as any other Cypriot.
In any case, who is native and who is not is irrelevant. All Cypriots should be equal, without racist or other discriminations, and that includes not only the native Cypriots, or those that came to Cyprus centuries ago, but even those that got the Cyprus citizenship yesterday.
Those that are are trying to divide the Cypriot people along ethnic lines hoping to have unfair and disproportional gains on the loss of the rest of Cypriots are only fooling themselves. They are playing the game of Turkey/UK and if they don't wake up the biggest losers of this game that is played against Cypriots will continue to be them.