That's an intriguing scenario to consider! If a one light year long stick were set in motion across the universe, and different parts of the stick were subjected to varying time dilation conditions, the speed at which the stick would appear to be traveling would depend on the perspective of an observer.
From the standpoint of an observer who is stationary relative to the stick, the entire length of the stick would indeed be moving at the same speed. This is because, in this reference frame, the stick and the observer are moving together, and the effects of time dilation are relative to their relative velocities.
However, if we were to consider an external observer who is not moving with the stick, the situation becomes more complex. Different parts of the stick would experience different time dilation effects based on their varying distances from massive objects or differences in relative velocities. Therefore, the external observer may perceive different parts of the stick to be moving at different speeds due to the time dilation effects.
It's worth noting that, for practical purposes, the differences in time dilation effects across a one light year long stick in the observable universe would likely be minimal. The gravitational and velocity variations over such vast distances are typically not significant enough to create noticeable time dilation differences.
In summary, while different parts of the stick may be subjected to varying time dilation conditions, an observer moving with the stick would perceive the entire length of the stick to be traveling at the same speed, whereas an external observer may perceive differences in the apparent speed of different parts of the stick due to time dilation effects.