Say I murdered someone on the California Nevada border and literally chopped up the body and spread it around the state line. How would it be decided who investigates the murder? Which police force?
If the actual murder takes place in Nevada, but the body is disposed of in California, Nevada will investigate with help from California. Nevada will prosecute for the more serious charge, California can then decide if they want to prosecute for the lighter. Federal agencies can be utilized for joint investigations, the degree of involvement is mostly dictated by the type of crime.
Now if the body was right along the line, the Feds would probably take lead, and then as information is developed, they'll pick which state it needs to be prosecuted in. Usually if they can't figure it by normal means, they'll choose the state where the major elements occurred.