I think Flemeth willingly gave her power to Solas, and agrees with what he believes he has to do. While she has been helpful to our protagonists in the past, she only helped them if they would do something in return for her. I don't think she is much different than Solas in her beliefs about what should happen in the future. If she did not want Solas to take her power, she would not have stayed by the eluvian in the end scene of the main game, as she knew he would come. She also indicates in Origins that she not only knows more than the Warden about a lot of things, but she also suggests that in a situation where the world is ending, she would lie to people to let them be happy, which is very similar to what Solas seems to be doing at the end of Trespasser.(1) She even hints at knowing about such a plot to destroy the world early in Origins, though I would guess that is a coincidence, as I don't think the writers were planning to make her Mythal that early in the series.(2)
As far as what her gift to Solas entails, I think that while we cannot be certain, there are a lot of things we can guess based on what has been said in Inquisition, previous games, and what happens in Trespasser. Solas most likely did not receive Mythal's soul, as there is no presence of the voices from the Well of Sorrows, and he doesn't take control of the Inquisitor when doing so would be to his advantage. While I realize that this could be simply because not everybody had their inquisitor drink from the Well, it wouldn't have taken much work to add a couple of lines. The souls of Mythal and Flemeth are connected, so if Solas didn't take Mythal's soul, he didn't take Flemeth's, either.(3) A developer's note found in Inquisition's code tells us that Flemeth sent her immortality to Morrigan through the eluvian. If Solas didn't absorb her soul, did she send it through the eluvian as well, or, as only a fragment of Mythal, did she have another vessel to use in case she died, as she did when Morrigan tried to kill her?(4)If he didn't take her immortality or soul, what did Solas get from Flemeth?
I have a lot of theories in mind right now, but most are too undeveloped for me to discuss very well yet, as I haven't even watched all of Trespasser or any of Descent. Just for fun, though, I'll tell you that my ideas include Flemeth using time travel and Sandal being an archdemon.
1. (The Warden approaches Flemeth after Morrigan asks him to kill her.)
Warden: I want the truth.
Flemeth: The truth! As if it were nothing! How like a man.
Flemeth: No, no. Far better the lie. Far better the comfort of blankets and shadows and a mother's love.
2. (The soon-to-be Warden, Alistair, and the two unfortunate men who later die in the joining go to Flemeth's hut to retrieve Grey Warden treaties.)
Barris: If she's really a witch, do you want to make her mad?
Flemeth: There's a smart lad. Sadly irrelevant to the larger scheme of things, but it is not I who decides. Believe what you will.
3.(Flemeth speaks about Mythal in her appearance in Inquisition.)
Flemeth: She is a part of me, no more separate than your heart from your chest.
4. (Flemeth explains herself to Hawke.)
Flemeth: Must I be in only one place? Bodies are such limiting things.
Flemeth: I am but a fragment cast adrift from the whole. A bit of flotsam to cling to in the storm.