How do I move someone on Medicaid, in skilled nursing care, to another facility in another state? What is the approval process for Medicaid?

Contact Medicaid. Looks like you have to terminate in one state and apply in another but I am sure it can be done so that coverage doesn't lapse.
Found this article that might be helpful; but ultimately contact Medicaid.
https://www.medicaidplanningassistance.org/transferring-medicaid/

I would talk to a Medicaid caseworker in the State you are transferring the person to. Be aware that Medicaid is a State thing. If you need to set up residency, this means that the person has no Medicaid even health insurance wise. And, as asked, how are you transporting them there.

The state pays a share of his care. Problem is that he has to move first to reapply for Medicaid. You might have to take him home, establish the person as a state resident, and provide 24 hour care on your own as you wait for acceptance. You cannot just find a Medicaid bed and have immediate acceptance. You have to wait for a bed to open up, or once accepted on Medicaid, the person goes to the first open bed. What if that Is miles away? Is this safe for discharge? How will this person be moved? Private plane? Ambulance? Who has the deep pockets to do this? Who is the person's legal POA. GOOD LUCK!