An arraignment date will be set by the Court, and the person charged will be required to appear. Given the nature of the charges, they will most likely be required to post a bail to avoid incarceration during the pendency of the case. If they fail to appear, the court will issue a bench warrant for their arrest, which can be the basis for their arrest in the other state. They would then be subject to extradition back to NJ to face the charges. This person needs to discuss this with an experienced NJ criminal defense lawyer before their court date. Good luck.
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Mr. Bailey is as always correct. I would also add that any DV will proceed without you and a PRO will be issued. The warrant will be on the computer. The local police in the other State may execute the warrant and then you will be held for NJ. You can post bail in most States of the fugitive warrant pending extradition but why go through all that. Speak to NJ counsel ASAP.
The case in NJ will continue regardless of what state the defendant resides. The defendant is required to appear for the different scheduled court events. If the defendant does not appear, a warrant for his/her arrest will issue, on which he/she can be arrested in the other state as well. If that happens, then that triggers bail and/or extradition issues. An experienced NJ attorney is necessary for such a situation.