In the U.S., in the event of a hot pursuit (i.e. It's a car chase, and the suspects are racing to the border Thelma and Louise Style) the pursuing Law Enforcement Officers are permitted to continue the pursuit out of their jurisdiction, until such time as LEO of proper jurisdiction can relieve the out of jurisdiction pursuit. This is true no matter if it's a state California Highway Patrol entering Arizona, San Bernadino Sheriff's Deputies entering Los Angeles County, or LAPD entering Compton (Compton is not in the City of Los Angeles, but rather the County of Los Angeles and is policed by the Los Angeles Sheriff's Department (LASD).).
In the event that charges are filed against someone who is in another state, they are extradited from one state to another. Typically this is a quick process, with the state requesting extradition going before the court in the state receiving extradition. The requesting state must produce documents of the indictment that have been signed by their state's chief magistrate of governor. Upon doing so, the receiving state will affect the arrest and have the suspect remanded to the custody of an agent of the requesting state.
Unlike international extraditions, all extraditions in the United States are backed by the full faith and credit of the United States, which means that in the above scenario, the act of breaking a California Law is grounds enough to justify the extradition, and there is no argument made to reciprocity (aka It's not illegal to do it in Arizona, so Arizona refuses to extradite). Failure to extradite can be grounds for a suit against the receiving state, which will be heard by SCOTUS.
As for your P.S. Borders are a funny place of existence, but a South Carolina police officer may arrest him just fine and prosecute him. In addition, because the suspect is committing a crime that crosses state borders, the feds can always have jurisdiction and can try the pot head regardless of how much he was crossed. This means that South Carolina and the Federal Government can seperately take a crack at him. Furthermore, possesion of Marijuana is still a federal crime, so if the feds wanted to, they can arrest Bob in North Carolina because the NC/SC border is still within U.S. Territory (The reason they don't is because in the grand scheme of things, the guy blazing it on the border of two states is not something the Feds want to waste their resources on. If Bob had enough to qualify for possession with intent to sell, that would make it a bit more interesting, as he's trafficking over state lines, which instantly puts it flatly in the DEA's jurisdiction (South Caroline would almost certainly be happy passing this to the Feds, because Federal Crimes are more serious and generally there's no time off for good behavior with the Feds.).