Character mistake: Kimble has just finished talking on the phone with his lawyer when the cops hear the noise in the background. They figure out the sound is that of an El-Train. They then start to list the different places that have an El. One of the cities they name is Milwaukee - there has never been an El-Train there.
Character mistake: After Kimble jumps from the drain pipe over the waterfalls, the scene goes to Gerard standing in the pipe telling his crew to leave, then they all leave without picking up Gerard's gun that Kimble threw down next to him.
Character mistake: The investigators say that the prints found in Sykes' apartment belong to Kimble after dusting for them, making it clear they didn't use any equipment for fingerprint analysis or testing. No forensic investigator would confirm who the prints belong to until they run a thorough analysis. It seems as if they were basing the fact that they belonged to Kimble because they traced the call to the apartment, rather than acknowledge that the prints could belong to someone else.
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