Can an agent for the FBI arrest someone for a local charge

Can an FBI agent arrest say someone in public for a crime enforced by the local police or state police?

For example, I saw on youtube where some men were pranking the public acting like identity TH. out side of Target. They came to this one man and asked," ok I got his info. Do you have a credit card in your back poket?" he replied with " no I have a badge with FBI printed on it." Did that agent now have the right to arrest the man, seeing as identity theft is a criminal charge. And the viewing personal info of a federal agent is a federal charge by the department of criminal justice.