Is there a relationship between the rise of the big piston and the fall of the small piston in the traditional hydraulic jack?

No, it is proportional to the area ratio of the big piston and the small piston. The stroke of the big piston is smaller than that of the small piston.

Every time the small piston is pressed down, the hydraulic oil will be squeezed into the large piston cavity under the internal pressure of the small piston cavity, and the volume of the hydraulic oil flowing in this process is the same, that is, A×L=a×l, then L=a×l÷A, L = (a/a )× L;

So: l/l = a/a.

Where: A, L large piston area and stroke, A, L small piston area and stroke.