Can you see the starting and the ending of a light beam passing in the distance? This question arose to me when I saw a SciFi movie where they shot with laser guns and you clearly could see the dashes of light beams travelling from the shooter to the target. Nonsense of course, reality is much more boring.

But I wondered if you would just need more distance: 


Given a limited light beam traveling from A to B, where "limited" means that there is a start and / or end of the emission source causing that beam.
Given enough distance from us to A, B and the line between them
And add some kind of nebular smoke in between just enough to disperse as much light as needed so we can see it here but not as much to prevent it from reaching B.


Couldn't we observe the starting and / or the ending of that light beam, progressing on it's way from A to B, like a jet trail in the sky?

If it is possible, where can this be observed? Or do the laws of physics forbid such a thing?
Nothing forbids this, and it is actually observed astronomically. You need a very bright source of light: such as a supernova (which isn't a beam, but radiates in all directions) and very large distances. The flash of light can be seen spreading out from supernova in a circle, as it illuminates the dust and gas ejected by the supernova progenitor star in the years before

The effect is known as a light echo. Over time the "beams" of light move further out, and so the echo widens, as shown in this series of images (from an Australian telescope so you have to read from the bottom up, 507 days means 507 days after the supernova in 1987.) For scale, 5 arcmin is quite small: The moon is about 30 arcmin. source