An across variable is a dynamic variable that is determined by measuring a difference of values between the two ends of an element.
Consider a damper being pulled from both ends.
If you measure absolute velocity at the left end of the damper, it is negative. If you measure absolute velocity at the right end, it is positive. The damper clearly does not transmit velocity from one end to the other, because the absolute velocity is observably different at the two ends.
To determine the lengthening velocity (v) of the damper, you have to subtract the two absolute velocities of its ends. You have to measure the relative velocity across the damper. Thus we say that the velocity of a damper is an across variable.
An across variable is in contrast to a through variable.