It is so confusing, I'm not sure how it works and hopefully someone with more legal knowledge than I can help to answer you question.
I do know, I have seen the FBI come in as consultants to help solve crimes. The same with the US Marshalls. They are allowed to cross state borders to investigate crimes that have taken place in different states. Although the FBI and US Marshalls have greater power to travel the US in investigating crimes, when it comes down to the court process, the charges need to be filed in the jurisdiction where the crime was committed.
That is why we are seeing so many different LE agencies involved in this case. Each Parrish or city plus county and state can all get involved for different reasons. Taking it across state lines I believe would then get the US Marshalls involved. FBI is brought in for advice, unless it ends up being something on a bigger spectrum such as organized crime.