What rights do the siblings of elderly have in their care?

Contact a lawyer who specializes in elder care. In most states, there is a hierarchy list of who is next of kin or can make decisions: spouse, followed by adult children, followed by other family members. You may need the lawyer to draft up official documents to "remind" your aunts/uncles that you are next of kin after his wife (your mom).

Eldestdaugh1: Something seems amiss here with two of his siblings becoming involved in his care, attempting to move him out of state and taking control of finances. Not to mention that the siblings of an elder are no doubt elderly themselves and perhaps may struggle with caregiving duties. Retain an elder law attorney.

Since he is no longer to manage the finances, the wife should be in control, and if the mother cannot, then you as the only child should have control.
His siblings have no right to anything jointly owned my mother and father

Hire an attorney yesterday.
Everything depends on who has legal authority AND the medical diagnosis of your loved one.

Gena / Touch Matters

I’m should think that his spouse would have responsibility since all their assets are shared. He needs spouse or you named medical and durable POA or guardian.

Just to be safe, please see an elder care attorney in your area who will know all the state laws.

In answer to your question about what rights siblings have in the care of the elderly, the simple answer is this:
Siblings of an elder have rights GIVEN THEM by an elder in so far as POA, Trustee of Trust, Guardianship goes. If the elder is too legally incapacitated to make decisions then the siblings have the rights given them by a court after an action for guardianship or conservatorship.
That covers the question itself.
You seem to be wondering why THEY, and not YOU are in charge currently.
In order to answer that I am afraid I have some..............

QUESTIONS!
1. Why is anyone other than your father trying to be "in charge" of his finances and where he moves?
2. Does your father suffer from dementia?
3. Does your father have an appointed POA?
4. How old are you?
5. What is the status of your Mother's health?
6. What is the status of your father's health?
7. Why is your mother not in charge of decisions regarding her and her husband's health and finances if you father is debilitated and unable to make decisions?
8. Is there a current guardianship of either mother/father/or both?
9. When you ask your father's siblings why they are taking charge of his affairs rather than allowing you as the only child to do so, what have they said to you?
10. Do you have a good relationship ongoing with your parents and with your father's family?
11. Do you live with your parents? If not, how long have you not.
12. Do your parents now require 24/7 care.

Answers to but a few of these questions might give us some minor clues as to what might be going on here.
Otherwise will just wish you the best of luck ongoing.

Tell them to stay in their lane.

I, personally, would see an attorney about how to protect the marital assets from the sisters.

According to law, your mom is his legal next of kin with you as the next in line. The sisters have no standing to legally swoop in and move him and take over finances.

What exactly are your dad's medical issues that would require intervention from anyone? Is he driving this or are the aunties interfering and pushing him? More information can get you better help.

What is "a multi-year medical struggle" and why is it relevant? How are F's siblings "involved in his care"? Without a bit more detail, this doesn't make much sense.

So the situation is Dad's Sisters want their brother living closer to them. Is that right? Why?

I am puzzled to why Sisters would think they can & should override any decisions made by Dad himself, his Wife (your Mom) & you?

Have these Aunts explained their concerns?

Are there family history influences hidden here? eg Protective much older sisters that raised him? Or they don't have good relationships with your Mother & yourself?

CONCERN is one thing isn't it... attempting to CONTROL is quite another.

Does anyone have POA for your father?
His next of kin would be his wife, your mom. After her would be his adult children.
Are both parents incompetent? Are you able/want to care for your father? If so and there is no POA, then you could see a certified elder attorney and file for guardianship.
Look up guardianship in your state and read the rules.
Your fathers estate would pay for the legal fees if you win.
Start with seeing an attorney to better understand all that is involved.
If the siblings have filed for guardianship and you received a notice from the courts you must respond quickly. If they have been caring for your father and his children have not, then chances are they will win. There are many unknown details that you should discuss with an attorney.

You are making an assumption that just because you're his daughter you "get" to be in charge of his decisions. Unless you -- or anyone else -- has been assigned as his PoA by him, and he is now cognitively or physically incapacitated, then only his PoA gets to act on his behalf. But if he isn't incapacitated, or has no PoA, then he is still making his own decisions. You have no power in this situation, but his siblings may not, either.

Are any of his siblings his PoA? Is he in his right mind so that he's voluntarily going? Or, he has no PoA and has cognitive impairment but no diagnosis?

There are not enough details from you. It seems your Mother is still alive and still married to him... is she unable to care for him or visa versa? Or are they both in need of care? What states are involved: your parents' home state, the siblings states? Is he in his home now or in rehab or a hospital or facility? What "attempting to move him" mean? What is going on with your Mom?