You need a cop, or better yet a criminal lawyer from the defense or prosecution side, or a criminal justice academic to lay out the basics for you, none of which am I.
How does a bad guy become a 'suspect'? If a cop sees him in the process of committing a crime, let's say running out of a store waving a gun at people, the cop can arrest him.
But if a cop is called to a store where the proprietor is dead on the floor with a bullet hole in him and the cash drawer empty, but nobody saw what happened, and the coroner declares death by homicide, one supposes that establishes a criminal case needing investigation.
Now begins a search for evidence or witnesses or both. Let's say no witnesses, nothing useful in the way of fingerprints, but maybe an ejected 9 mm shell on the floor. So maybe that's the only clue, and the investigation goes off to check out that angle. But nothing pays off, and it becomes after a while a cold case. Nobody knows who did it, or where he or she may be--in town, out of town, out of state, out of country. No reason to think federal law has been violated, so feds are not involved.
But, maybe months or years later some guy gets picked up for possession, and in the process of bargaining for mercy, tells the cops, yeah, I know who killed that store-owner, it was Butch, he bought some shit off me and told me he got the money by robbing that store and he had to shoot the guy. So now the cops report their findings to the DA, DA goes to court, gets a warrant, APB goes out.
Police out of state months or years later stop Butch for traffic violation, call in his details, data search turns up warrant, Butch gets cuffed and taken in. Police are notified in jurisdiction where warrant originated. They get an extradition order, and Butch is then brought back to be questioned. Did you kill that storeowner?
No. I dunno nuttin bout dat. -- You're lying! We know you did it! -- I want a lawyer!
Police give details to DA. DA takes Butch in for arraignment, judge finds probable cause, Butch held for trial, bail denied. Public Defender tells Butch his best option is take a plea deal, eligible for parole after thirty years, otherwise death by whatever.
OR, somebody's mother in NYC goes to police station, says her daughter didn't come home last night, but phoned to say her pimp was going to hurt her. Cops get this all the time, don't give a shit. Mom wants to file complaint against pimp, cops tell her no evidence, no complaint. Eventually missing person report filed.
Two years later rotten corpse discovered in NJ. By some miracle, identity is made, mom is notified, she curses cops for not saving her daughter when she first complained. Coroner says homicide. Now NYC and NJ both open cases, but it's primarily a NJ case, because that's where the body is, presumably where crime was committed. Pimp is the only 'person of interest,' based on mom's hearsay, but no material evidence, no witnesses, just mom's story.
BUT one of pimp's other floozies saw the killing, fled to Maine, fearing pimp will try to snatch her and use her for his next snuff film.
Years later, Floozie spots pimp before he sees her. She goes to local cops for protection, tells how she saw pimp kill girl, gives names, dates, NYC places. Maine cops send details to NYC, now NYC DA has a case, has a witness, gets a warrant, asks Maine to be on the lookout for pimp, please apprehend and hold for extradition. Feds are notified of possible interstate sex crime, so federal case is opened. Both NYC and feds want pimp for questioning; NJ is interested, but on standby now that witness puts the crime in NYC. Feds can intervene in Maine, NYC has to depend on cooperation. Both Maine cops and feds have power to apprehend. How it gets sorted out from there, assuming pimp is caught, is beyond my 'expertise'.
This is mainly guess-work on my part. No charge for this advice.