A through variable is a dynamic variable that has the same value at both ends (or any point) of an element.

Consider the figure below:

-->v1 -->v2
| ________ |
| |
f1--->| damper |--->f2
|________|

Vdamper = v2 - v1

Apply Newton's Law to form the equation

f1 + f2 = ma

By definition, we take the mass of the damper to be zero, so

f2 = - f1

Now define f2 = -f1 = F

We have the figure:

-->v1 -->v2
| ________ |
| |
F<----| damper |--->F
|________|

We can say that the damper transmits a force, F, through itself, end-to-end, unchanged. Thus we say that force is a through variable.

This is an underlying concept in forming free body diagrams in physics

A through variable is in contrast to an across variable.