Why did they move?

The Pull Factors 2

When the ultimate defeat and surrender of the Circassians become apparent the Russian Government made overtures to the Porte, in view of ascertaining whether the Sultan would receive into his dominions such proportion of the Circassian tribes as would desire to leave their country. The Turkish Government consented to receive them on the condition that the emigration should be gradual and should not commence before the fine season. It was then believed that 40,000 or 50,000 would avail themselves of this refuge. The progress of events, however, has been so rapid that these stipulations have been totally disregarded, and the successive victories of the Russian army in the Caucasus produced the panic and the flight which have been recorded above.

Source: The Times of London, 9 May, 1864


The evidence suggests that the Ottoman administration made a commitment to take the Circassian exodus before they had a clear picture of just how many migrants would seek refuge within the Empire, or how urgent would be the demand for transportation away from the Caucasian shore.