Why do different parts of a spring with mass stretch by different amounts?

If a massed spring is kept horizontal why should different parts stretch by different amounts?

Suppose a one end is fixed, and the other is free
with a velocity $v$ (uniform velocity)

Suppose in $t$ seconds the free end moves a distance $x $, should the other elements move a distance $x$ along with it?

Also I have seen some stack exchange answers (of similar questions)already but I couldn’t understand them

Can anyone provide a mathematical proof also ?