I know there was a non-unanimous vote, but did it work via everyone voting or just a few chosen country? Also, how did expulsions work? Was there a vote for that?
Article I of the Covenant of the League of Nations included
"Any fully self-governing State, Dominion or Colony not named in the Annex may become a Member of the League if its admission is agreed to by two-thirds of the Assembly, provided that it shall give effective guarantees of its sincere intention to observe its international obligations, and shall accept such regulations as may be prescribed by the League in regard to its military, naval and air forces and armaments."
The Assembly under Article III consisted of representatives from the members of the League, as opposed to the Council which was a smaller body somewhat like the existing UN Security Council.
Assumedly, during an Assembly meeting, any prospective membership would be put up for a vote to the League as a whole, rather than its most important members.