So something that most French Revolution books I've read seem to gloss over is the election of the initial third estate.
All that most books seem to say is that only propertied men could vote in it. However, given the political atmosphere, there must have must be campaigning and conflict across the entire nation. I'm really struggling to find any sources. Unfortunately, I can't really read French so most of the pamphlets of the era are out of bounds to me.
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Edit*: to expand on this, lots of the members of the third estate were prominent citizens like: Bailly, mayor of Paris; and Sieyes, who authored the pamphlet "What is the 3rd Estate".
There were 44 Parisian deputies, surely they can't have run unelected? Also how did they vote for these people?
I can answer more targeted questions if you'd like, but it sounds like what you're looking for is Becoming a Revolutionary by Timothy Tackett! It is ALL about the National Assembly, from the run up to it through until it's end. I absolutely love the book even as a pleasure read, Tackett nails a writing style that is formal and informative but still fun and interesting (and very accessible). I'd highly recommend checking it out, since it sounds like just what you're searching for!
It talks about the prominent men and also how they felt about campaigning (it was frowned upon) and some of the elections.
As to the "how" it was very dependent from province to province. I know in Arras for example it was a week-long, drawn out affair that turned quite bitter. Many of the men who had come in from rural areas were upset since the elections took so long they had to return before casting their vote. In provinces like Brittany there was armed conflicts, whereas Paris was such a confused mess that it took until weeks after the Estates General had begun to sort out their election rolls. Like many things in the French Revolution, there wasn't one way, it entirely depended on the area.