Here is a description of the motion of two springs in series. The premise is that the force on the two springs is the same.
This is derived from the following reasoning: when I pull the mass with a certain force $F$ at some point I reach the equilibrium position.
So every piece is stationary and in particular the point between the springs is stationary: we conclude that the forces applied in that point (the two elastic forces) sum up to zero. We derive that the elastic forces are equal and they're equal to $F$.
The problem is: this is valid in the equilibrium position. I can't understand why this conclusion is extended for every position in order to derive the equation of motion: