Yes, time dilation affects moving objects. In fact, time dilation is a fundamental consequence of the theory of special relativity, which applies to all objects in motion relative to each other.
According to special relativity, when two objects are in relative motion, their measurements of time intervals will differ. An observer who is moving relative to another object will perceive the other object's clock to be ticking slower compared to their own. This effect is known as time dilation.
To put it simply, if you were traveling in a spaceship at a significant fraction of the speed of light relative to a stationary observer, time would appear to pass slower for you compared to the observer. From your perspective, your own clock would be ticking normally, but you would observe the stationary observer's clock to be ticking slower.
So, time dilation affects all objects in motion, regardless of whether they are stationary or moving. It is a consequence of the fundamental principles of special relativity, which state that the laws of physics are the same for all observers in uniform motion.