Can a police department enter a neighboring state and process a crime scene?

I believe the answer is no, but this is just from general knowledge. As far as I know, as soon as the crime scene is in a different state, it is not in any other state's jurisdiction, meaning they would have no authority there. Even if a crime is commited in one state and then the criminal goes to another state, the original state's police can't do anything about it outside of the state borders.