Does Medicare/Medicaid cover costs to transport a patient with dementia from one nursing home to another located in a separate state?

My 69-year-old cousin in December 2013 suffered a heart attack in the lobby of her Miami condominium building hitting her head on the granite floor tiles as she collapsed. Evidently, she stopped breathing until the paramedics arrived more than five minutes later. She survived with hypoxia and has been in a nursing home ever since. Her conversational capabilities have dramatically improved along with her incontinence issues, but not completely. My cousin must wear diapers as she often "forgets" to rush to a bathroom when in need. She cannot give up on her belief that her mother is deceased since 2011 as she believes her mother continues to reside in her Miami condo or lives upstairs in the nursing home. Her boyfriend allows her to leave phone messages for her mother at her "condo" (unsure if that has any long-term effects on my cousin's recovery despite everyone explaining that her mother along with all aunts and uncles are no longer alive. My cousin's boyfriend even showed her picture's of the family gravesite in New York to no avail). My cousin is not 100% stable on her feet having fallen recently in her room in the nursing home. She spent two weeks in the hospital due to her hitting her head as she fell, the last week solely for occupational, speech, and physical therapy of which she receives little, if any, at the nursing home. Her boyfriend of 25 years has Power of Attorney (POA). They have lived together for several years in Miami. However, he is thinking of relocating to his co-op apartment in New York City wishing to move my cousin to a nursing home there. He hears this can be quite expensive. My cousin currently has Medicare, Medicare Supplement United HealthCare through AARP, and Medicaid. So, here are my questions:
1. Would Medicare and/or Medicaid cover the costs of transportation between state and, if so, what conveyances would be covered, i.e., commercial airline, medical aircraft, etc. or would the POA boyfriend be responsible for all costs?
2. If Medicare/Medicaid does cover such transport costs, would any medical staff be able to accompany my cousin during transport whether it be on a commercial airline or medical aircraft? The cost of time and a hotel in New York City for the nurse who accompanies my cousin on the trip as well as the flight for the nurse to return to Miami is a monetary factor, too. Would these be covered by Medicare/Medicaid?
3. My cousin's POA boyfriend is also considering moving my cousin into his New York City apartment rather than taking her to a nursing home. Would Medicare/Medicaid cover medical attendees coming to their residence, i.e., nurses, therapists, medical equipment (including special locks on the apartment doors to prevent my cousin from escaping for a walk alone which she has done on prior occasions) that she may be in need of?
4. Does Medicare/Medicaid offer some sort of relief for caregivers? For example, could a nurse stay with my cousin for, say, four hours a day to give her POA boyfriend an opportunity to get away to take care of his personal life needs?
5. My cousin's boyfriend is under quite a bit of stress sitting with my cousin each day for approximately 5-8 hours per day in addition to his own appointments for doctors, life needs, etc. He has been remarkable in caring for my cousin. His fear is that he will die before my cousin's eventual demise. In such a case, I would have to relocate my cousin to my home city in Palm Springs, California then becoming her caregiver. Would Medicare/Medicaid cover her daily costs, including the transportation to California? Would they pay for my travel to either Florida or New York so I may collect my cousin to bring her to California?

Thank you for your attention to my questions.