Can my mom unilaterally decide to move my grandma (who is in good health) out of a nursing home without consulting the rest of the family?

I'm writing because I am not sure where else to turn and hope that some people who have been in similar situations can chime in with their thoughts.

Here are the details:
My 83 yr old grandmother has been in a nursing home in rural Ohio for 7 years. She is by far the healthiest person in the facility. Her daughter (my mother) has been the one making sure she's been taken care of for many years now (POA, Healthcare Proxy, etc.). My mother is also the only family member within an hour of grandma. Grandma has four grandchildren, 2 around an hour away that she sees a couple of times a year, 1 in D.C., and myself in NYC.

Here's the issue at hand:

Part #1: My mother has decided to move to Texas and bring my grandma with her. The plan is that they will live together down there in an apartment until the time comes when grandma actually needs to be in a nursing home. My grandma, as far as I know, wants to go with her on the big move. My grandma has no retirement savings or assets and only gets a social security check every month (which currently pays for her nursing home fees). My mother is on unemployment and looking for work in Texas for when she arrives. Basically, they each have a limited income but when pooled together it will allow them to pay for rent, etc. in Texas. Although they'll be poor, I'm confident they'll make it.

Part #2: I am moving with my wife and 2 year old son to a house 4 miles away from where my mom and grandma are planning to move. I will be moving there around two weeks prior to their expected arrival.

Part #3: My mother didn't tell my sister or cousins that she planned on moving grandma out of the nursing home and to Texas with her. She only talked to me about it before making all of her plans. This caused quite a few hurt feelings when the news was received secondhand. Three grandchildren (my sister and 2 cousins) are really upset and think that my mom isn't in her right mind for wanting to bring grandma to Texas. They feel that grandma is just fine where she is and that it's alright if my mom goes alone. They think that because my mom wouldn't financially be able to survive in Texas without sharing expenses with my grandma, she is taking advantage of grandma's limited amount of money and strong arming her into moving out of a safe environment.

I've been playing the middle man in all of this. While I understand everyone's concerns, I think that if grandma is healthy enough and sincerely wants to move with mom to Texas that she should go. The three other grandchildren are so upset that they say that if mom doesn't agree to put off her plans, they will take legal action to try to stop her. I don't know what they could try to do. I guess that they would have to prove that my mother doesn't have her mother's best interest in mind, or that she's putting her in danger?

I understand that they want to make sure grandma is safe, but to put my grandma through a painful legal battle just because they don't agree with a choice seems like a serious overreaction on their part.

I guess I would just like to know if my other family members could legally put a stop to this. It doesn't appear to me that any wrong-doing has taken place.

Can my mom legally make a unilateral decision about my grandma's living situation (with my grandma's consent) without consulting all the grandchildren?

Any experience, thoughts, or suggestions?