There were obviously no elections

Oh boy! There were elections and so much of them!

I will answer in the cadre of the last États Généraux, those of 1789. Since their instauration by Philip the Fair, France had know about 30 États Généraux, but the precedent was in 1614. The rules for each of them were complicated and different.

So in 1789 the Prime Minister Necker under the pressure from the Tiers État had to empower them, doubling its size. All people who paid a direct taxe, was 25 of age and was a resident were allowed to vote. Two deputies were to be elected in the country for every parish or commune with hundred fires or less, three for those with three hundred fires, etc. Then those deputies would go with their grievances to the next round of elections in the baillage or the sénéchaussé secondary or principal, where they would select two of them to go to to the next round with a synthetic grievance notebook. In the cities the votes where often organized internally to the corporations (like the bakers' deputies) or the district, and then you would have another round or more in order to select only two deputies for each selected area.

But that's the gist of it because at the time France was not neatly delimited in areas with the same laws and customs, many area had some particularities inherited from past compromises or special privileges.

Also you should note that you could not be candidate to the deputation: it was expected that everyone should vote in accordance to his intimate choice, so announcing a candidacy was clearly corruption! You could only do that in secrecy or in disguise. The entries, allocutions, mass, meal times where the manner to get some spotlight and make people believe you would be a good deputy.

How would you vote? The votes were conducted in diverse manners: often collectively by seating of standing up, by raising hat or by moving in the place; when you had some risks of conflict you could do it secretly, the illiterate would vote by telling their votes to the election board (president, secretary, three scrutineers it was often composed by the community elders), the board was also charged of writing the proceedings and resolutions.

They were also some funny things like Guillaume de Bonnemant a deputy of Arles (in southern France), who sieged illegally after an error of the comity in charge of the verifications. His city ought to have only three deputies, he was the fourth. Some areas were not mentioned in the royal statut but nevertheless shipped some deputies, like many overseas territories (often colons residing in Paris), and were authorized to sit in the assembly. On a sad note the deputies form La Réunion died en route in a shipwreck.

That was a complicated and lengthy process but it did attain its objectives at the end the Tier États had its 578 deputies. The selection was drastic and the majority of the deputies were men of the law (lawyers, prosecutor, judge), often relatively wealthy and urban, not poor farmer or cobbler.