Why did they move?

Why did so many Circassians go to the Ottoman Empire? The Pull Factors

The Turkish Government is willing and anxious to receive the fugitives, and incorporate them into their own population, but the movement has been so sudden and so extensive that it bas been impossible to make provision for the hosts that are daily pouring in. It is calculated that no less than 300,000 will, within the next two or three months, seek shelter in this country, and half that number are now seeking the means of transporting themselves to the Turkish coast of the Black Sea.

Source: The Times of London, 13 June, 1864


The first pull factor here was that the administration of the Ottoman Empire reacted very quickly to the emerging situation in the Caucasus and made it clear that they would offer refuge to Circassians.