Does the hospital have the right to move a critically ill patient due to their mistake in paperwork? What can the family do?

Not sure what you are asking. The hospital has jurisdiction over their view of stabilizing patient and moving to another facility they see as appropriate to deal with moms current condition or needs. If patient has signed HIPPA or filed a HIPPA with dr for them to be able to contact family or share health info, they may do so; if not, they are under no obligation to try to find and contact family members.

If you are questioning her care or hospital mistake in her care, then speak with hospital to collect details and file a grievance or contact an attorney to file a grievance.

If you have DPOA or medical POA you can make some decisions with regard to further care and where she is taken.