Can a person transfer a DUI case if he moves to another state?

I assume you mean transfer an ongoing case to another state.  The only court that has jurisdiction over the case is the state where it accorded. Wherefore, the state that charged you with the DUI has to be the state that resolves it.

However, you can deal with the DUI in another state without ever going back to that state.  Many people get arrested while traveling through a state and never come back.  Usually, it requires filing a motion to excuse your presence or to allow you to appear telephonically.

Lets say you accept a plea to a reduced charge.  The court will usually allow you to plea by affidavit.  That means, you will have to send payments of fines and proof that you completed classes to the court that has jurisdiction.  But, you never have to go back.

Hope this helps.