How Turkey and Greece exchanged their populations?

In 1923-24, Turkey and Greece exchanged their Orthodox and Muslim populations, respectively, using not ethnicity or self-identification but religious marker as the criterion for the exchange

Subsequently, hundreds of thousands of people were uprooted from their homes and sent off across the Aegean Sea. While it was based on religion, the exchange ended up sending Turkish-speaking Christians to Greece and Greek-speaking Muslims to Turkey

Not only Cretan Muslims but also similarly Greek-speaking Ioannina Muslims were sent to Turkey during the "population exchange." The exchange was executed according to religion, which meant that Ankara, in return, sent Turkish-speaking Orthodox Christians of Cappadocia to Greece

From a cultural vantage point, my "favorite group” exchanged between Turkey and Greece is the Romance-speaking Muslim Vlach community of Greece. Believed to descend from Roman colonizers in the Balkans, (Muslim) Vlachs are the perfect Venn diagram of the Roman and Ottoman Empires

Greek Orthodox populations of Istanbul and Aegean islands in Turkey and Muslims of Western Thrace in Greece were mutually exempted from the religiously-determined "population exchange," a concession to which Ankara agreed so the Patriarch of Constantinople could stay in Istanbul

So were Muslim Albanians of Chameria in Greece and Orthodox Arabs in Turkey--whilst Athens and Ankara wanted to "exchange" them. Even homogeneity-obsessed interwar League of Nations diplomats couldn't fathom sending Arabs from Turkey to Greece and Albanians from Greece to Turkey.