In my cat sprite I had two costumes cat1, cat2. In the cat script I used a block to set the costume to cat1.
When I dragged the cat script to the dog sprite, the set costume block still said to set the costume to cat1 - which does not make sense in the context of the dog sprite. Furthermore, there was no way to fix it other than replace the block.

Actually, it seems that you can fix the sprite without replacing it with a new set costume block. I cannot currently reproduce the behavior where clicking the dropdown did not show the appropriate costumes for the current sprite.