Revealing mistake: During the St. Patrick's Day parade, Kimble removes his coat. In the next scene, he hasn't got the coat anymore, and he HASN'T dropped it. Don't be fooled. If you look to his right in the scene just after he takes it off, you can see a man carrying a coat that is the exact same colour and design. I assume this guy is supposed to be hiding the coat from view, but you can plainly see it. The guy is wearing a green hat. Kimble had no time to give it to him before the scenes changed.
Revealing mistake: When Kimble jumps from the dam, the falling body is clearly a dummy, otherwise he broke every single bone in his body before he even jumped.
Revealing mistake: Just after the opening murder scene, there's a shot of a Chicago Police van responding with blue lights flashing. If you look closely at the roof lights between the beacon on the right side of the roof you can see a sign "FILM2" stuck to the bar. (00:01:55)
Answer: It's a cold compress. You squeeze it, and the inner bag breaks mixing chemicals and it gets very cold. It helps to minimize swelling. He gave it to Kimball for all the bruises he had.
Grumpy Scot