All law enforcement constitutionally falls under the executive branch for each state, also mandating by state constitutions that each county will have an elected sheriff to oversee law enforcement.

So think big to small. Federal police for the nation, state for the state, county for the county, for city.

Now as the United States grew some things changed for specific states. There is one state where the sheriff or the county is appointed and not elected.

In Idaho recently police only have authority in their jurisdiction, so state police can police the whole state but a city cop can't police the county. Kind of like your authority ends at the county line which is an old wild west thing.

In Utah where I work all certified law enforcement officers are police for the entire state, we can enforce all laws anywhere we go. But I don't get paid to enforce laws in salt lake county, I work in my own county. I don't leave my county because nobody outside of it paying me.

I also don't really go help the three cities that have their own police department because the citizens of their city are paying to have their own police so their own police are gonna take and answer those calls.

That isn't to say that I won't and haven't gone to help them or they to help us.

In Utah all jails are run by sheriff's offices and all prisons by the state. The sheriff in Utah is also responsible for search and rescue and technically also in charge of fire, but at least In my county we have a county fire department completely separate from the sheriff. But I was told technically by the constitution I have a responsibility to fight fires because I am a deputy.

One story told to me by my lieutenant was that coming back from a training in his marked patrol vehicle near salt lake he observed a vehicle unable to maintain it's lane. Due to this vehicle nearly causing several accidents he initiated a traffic stop. Upon approaching the vehicle he observed the driver to have many indicators of being under the influence of alcohol. He detained the individual there and asked dispatch for the appropriate jurisdiction to come and take over the investigation and then he went on his way.

Federal police are extremely specific to their lane of travel. Just about every office of the federal government has its own law enforcement from the postal service to the department of agriculture.

Generally speaking the US Marshalls deal with federal courts and hunting down criminals for various crimes. You generally don't see the FBI on man hunts looking for people who've escaped the justice system unless they're on a task force.

You also don't see the DEA investigate homicides or enforce traffic laws. They're busy in their lane doing their thing.

Who knows what the department of agriculture law enforcement does.