Can state courts give immunity? I would assume yes. What happens if a state court grants immunity to get at the evidence or testimony but the evidence or testim

Can state courts give immunity? I would assume yes. What happens if a state court grants immunity to get at the evidence or testimony but the evidence or testimony could lead to federal or other jurisdictional (another state?) charges or prosecution? How can one be forced in one court to be exposed in others? It would seem the immunity being granted would have to be universal. How can one court/jurisdiction grant such immunity beyond its jurisdiction?