According to big bang cosmology, we should see galaxies in an earlier evolutionary phase as they are more distant: is this indeed observed, do (very) distant galaxies or galaxy clusters systematically look different (except for a larger redshift) and, if so, what are the differences with nearby galaxies?
Yes, they do seem to look systematically different. As CuriousOne mentioned in a comment, a good place to start is the Wikipedia article on the various Hubble ultra-deep-field observations, which used very long exposures to image very faint and distant objects. This includes images of galaxies from less than a billion years after the big bang.
From this summary paper you'll find these two quotes.
From the abstract:
Visual inspection of the images shows few if any galaxies at redshifts greater than ∼ 4 that resemble present day spiral or elliptical galaxies. The image reinforces the conclusion from the original Hubble Deep Field that galaxies evolved strongly during the first few billion years in the infancy of the universe.
From the summary:
As found in previous observations with the Hubble Space Telescope, galaxies at these high redshifts are smaller and less symmetric in shape than galaxies at lower redshifts. These results confirm the conclusion of the first Hubble Deep Field observations that typical galaxies in the early universe look markedly different than galaxies today, showing that galaxies evolved rapidly in the first few billion years after the Big Bang.
I am sure there is a lot more information out there, and I apologise for the screen-scraping nature of this answer.