A few things to remember:
1. Flemeth and Mythal are one being and have been since Mythal "possessed" her. They can no longer be seperated, this is being said both in DA:O as well as DA:I.
2. Flemeth has more than one body. We know she had at least three versions of herself running around Thedas: One in the Korcari Wilds, one which saved Hawke from the Darkspawn and the one in the amulet which was carried to Kirkwall.
3. Possession/soul transfer requires both participants to be willing. Kieran's actions seem to prove that.
4. There are at 3 souls in play here: Flemeth/Mythal, Urthemiel and Solas. Either the Dread Wolf is just an elf, has taken the shape of one or has taken a host similiarly to Mythal, so their souls became one... so in the end, he's just one soul.
Let's get the obvious logic problem out of the way first: If possession means the souls combine, then Kieran shouldn't have been able to give up Urthemiel's soul. But the Dark Ritual worked precisely because the just conceived child didn't have a soul yet, so there was nothing there to combine with. Kieran grew up with his own soul, being a vessel for Urthemiel but never interacting with that soul beyond hearing voices.
To understand what happened in the final scene one needs to have a look at the Well of Sorrows.
https://www.youtube....9VxyQr30Q#t=997
Note the dark fog effect and the figure emerging from the water. That's Mythal. She leaps into the Eluvian.
Shortly after Kieran is being called into the Eluvian at Skyhold, he enters the Fade and there meets Flemeth/Mythal. He gives Urthemiel's soul to Flemeth, who takes it into herself. Unless she just changed the rules of how possession works, Urthemiel joined Flemeth and Mythal, three became one.
In my playthrough, I had Morrigan drink from the Well, she is now bound to Mythal. And here we see how tricky those immortals can be.
We were told Flemeth had many daughters through the ages and that she took over their bodies to stay young. What if that is only half the truth? What if that is the secret behind Flemeth having multiple bodies? Remember the amulet? The Flemeth who emerged from it said she was only a piece, a small piece. "I am but a fragment cast adrift from the whole."
How do you synchronize all the memories from your bodies? You put them into the Well of Sorrows and extract them from it later.
We saw that Mythal was -in- the Well of Sorrows. My theory is that Flemeth brought her daughters through the Eluvian to the Well of Sorrows. The sentinels would all be asleep and wouldn't notice. The daughters drink from the well, gain all the knowledge stored inside and bind themselves to Flemeth in the process. They then have access to the memories and personality of Flemeth/Mythal and can impersonate her but they do not possess her soul. Even just that would give them power (knowledge), as shown by Morrigan, who was able to turn into a dragon just like her mother after drinking from the Well. They also can't replace her mother since they are bound to her.
The real Flemeth might take over the body of a daughter if she needed a younger body, but that daughter would accept it willingly and their souls would merge. It is why she raises them to be just like her, to make the transition easier. Otherwise her personality would shift each time their souls merged. Or perhaps she likes them to have similiar personalities so they use the memories properly.
I think that the first Flemeth, the one who got possessed by Mythal, raised daughters, took them to the Well and went into it herself, becoming one with it. Her daughters then drank from the Well, turned into copies of her and went out in the world.
The real one, with the soul of Mythal, has been in the well ever since. All we met in the games so far have been copies. That also gives her the ability to choose which memories are shared. At the end of their life the daughters come back, share their memories and new daughters take over.
What does all that mean for the final scene? Everything and nothing. From the fog effect we can conclude that all souls have been transferred to Solas, he is now the vessel for Flemeth, Mythal, Urthemiel and himself. All four souls joined into one. At this point it doesn't matter if the Dread Wolf took over Mythal or the other way around, the end result only differs in the body they use now. That is why he says, "The people need me." I interpret it as "Please let us use my body."
There is no force involved, no struggle. Flemeth was evidently more powerful than Solas, he couldn't have stolen her soul or taken her over anyway.
As I said earlier, immortals are tricky. Flemeth probably still has other bodies running around in Thedas but now the Well of Sorrows is no more. To get the power and memories of her copies, Solas/Flemeth would have to possess them. And just like that the real Flemeth would be back.
P.S. Yeah, I know this theory has one or two holes. I give it maybe a probability of 20% to be true... but isn't it a clever little theory? It would fit Flemeth's M.O. perfectly. ^^