If we look into a far off distance at an object travelling towards Earth, shouldn’t we be able to see both the light from the object at some time t plus the light at some later time (t + delta t)?

Let’s also assume that the object is traveling fast enough that it is discernable. This point might be moot, since I’m not sure if such a situation is possible. I know that Rayleigh’s criterion could give us a lower bound for how far the images of the object has to be, though I’m not sure how complicated it would be to throw redshift into the mix.

This seems like one of those “Whoa this feels see weird causally but it’s just a natural consequence of things we’ve observed thus has little repercussions as to what limitations physicists actually work around.” Actually, I could see perhaps long exposure photos (or the telescope equivalent, if it exists) could run into issues.