According to our current understanding of physics and the limitations imposed by the speed of light, the other end of a 2-light-year-long stick would not move simultaneously with the end you are pulling.
The information about any change or movement you make on one end of the stick cannot propagate instantaneously to the other end. This is because information can only travel at or below the speed of light. Even if you were to pull one end of the stick, the information about that action would need to travel through the stick to reach the other end.
Considering that light travels at a finite speed, it would take at least 2 years for the information of your action to reach the other end of the stick. As a result, there would be a noticeable delay between the time you start pulling the stick and the time the other end begins to move.
It's important to note that the concept of a 2-light-year-long stick itself is purely hypothetical. In reality, such a long object would be subject to various physical constraints and likely not feasible to construct. However, even if we imagine such a scenario, the delay in information propagation would still apply based on our current understanding of physics.