She might very well snatch the kids, and run off with them and Vronsky to Paris.
I assume they didn't speak English but they did speak French. Therefore the choice of Paris over London or America.
I'm not saying it would be right, but I could see it happening -- at least in the modern world.
Oh, Inspector Morse! What big blue eyes you have...
In the novel she went to Italy with Vronsky and their Daughter, but then returned to Russia as they had difficulty making friends.
sharedon't forget she would be getting alimony too
so many movies, so little time
I don't think she'd have necessarily have to snatch the kids and run, as, after all, Karenin tried to divorce her twice. The second time he even offered to perjure himself in court by saying that he was the one who committed adultery; thus meaning that Anna could have the kids and was free to marry again while he wasn't permitted to by law.
Anna's choice to take only her daughter and go live as a fallen woman was born of both spite and guilt.
So, in modern day, Karenin probably would have instantly divorced her the first time he wanted to with no one to talk him out of it for Anna's sake. Annie would have still been in the womb at that point, so he'd never have to see and get attached to her. Instead, she'd have been all Vronksy's legally as well as biologically, and Seryozha would probably have seen his father on weekends and alternate holidays.
Oh, a modern woman would bang Vronsky and feel guilty about it, start divorce procedings and plan to marry him, realize it'd never work because he was a cheating douche who was already onto his next conquest, and try to stop the divorce and go back to her boring, comfortable circumstances. But Karenin would have had it with her by then.
Of course someone could get a novel out of that, but the ending would lack the drama of the 19th century version.
Both of Anna's men were too good for her. She didn't seem to care about her daughter at all, and her son not enough to avoid the affair or its consequences.
Kerenin her husband was honorable, and even offered her a way out of the marriage that would have allowed her to marry Vronsky by taking the blame for the marriage's failure on himself. In the end, it is he that raises both of Anna's children.
She refused to feel any shame for the situation she created, and nearly drove her lover insane with her insecurity and jealousy. She was a self-centered immature twit. I have a feeling that even with the sorrow he must have felt, there must have been also a sense of relief. He, on the other hand seemed not to have any feeling for his daughter, but there is no doubt that he loved Anna. He never marries and goes off to war.
There is a Russian version of the story on YouTube called "Vronsky's Story" that tells the tale from his point of view in his future in flashbacks after her death. It has English subtitles. The actors playing Anna and Vronsky are married in real life, are very attractive, and have great chemistry. It is much better than this trash.