Why does Eckels compare people to chess pieces?

After leaving the Path and stepping on the prehistoric butterfly, which dramatically alters the entire course of human civilization and results in Deutscher's presidency, Eckels and the group of hunters return home from their Time Safari expedition. Upon their return, Eckels immediately notices that something is awry by the subtle change in the atmosphere and environment of the Time Safari office. As Eckels acknowledges the subtle changes, Bradbury writes,

He could feel them [the people] moving there, beyond the walls, almost, like so many chess pieces blown in a dry wind...(9).

The narrator compares the people, who have been unknowingly affected by Eckels's mistake, to chess pieces being blown in a dry wind in order to emphasize how their fate is predetermined and out of their control. Similar to chess pieces, which are controlled by an outside force and have deliberate movements, the citizens in Bradbury's future society are not in control of their own fate. The fact that Eckels has the ability to alter the entire course of human history, which invariably affects the lives of millions of citizens, corresponds to the function of chess pieces.

Eckels doesn't compare people to chess pieces directly. However, there is a strong sense, conveyed  by their safari guide to the past, Travis, that people must behave with the rigid precision of chess pieces during their time in the dinosaur age. They must not disturb one bit of nature. As Travis explains, they must, like chess pieces, stay strictly in their own "square." In this case, it is a specially constructed path that keeps them from contact with the jungle. Eckels says to Travis:

Then it wouldn't pay for us even to touch the grass?

Travis agrees and tells Eckels and the group that even stepping on one mouse could have dire effects on the course of history. The mouse would have no more children, and this could mean a fox that depended on eating mice would starve, and if enough foxes starved, a lion could as well. Whole civilizations could conceivably not arise.