Nothing. The police will do research on cold cases. Prosecuting attorneys are fungible.
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The case stays at the DA's office, but managed by another DA. The new DA may take a fresh look at the case, or might have less interest in the case. If you're the suspect, keep out of trouble. If you're the victim, or family member of one, you can check w/ DA as to the status of the case.
The case is not simply dropped because the prosecuting attorney is transferred. Another attorney usually picks it up and continues the work. However, I agree with Ms. Hearnes' answer. If you are the victim in the case and are seeking to have the case move forward, contact the DA's office and find speak with the supervisor to make sure that the case isn't simply ignored. If you are the potential defendant, then lay low and count your blessings.