Good question!

The process of appointing elders varies among denominations and individual churches. I have witnessed a couple of ways this is done:
1. Consistory recommends candidates to the congregation who then vote for the man/men of their choice. The consistory then approves (typically) and ordains the winner(s).
2. Congregation recommends candidates to the session which initially examines them. Then the candidate(s) are examined/vetted by a special committee of the Presbytery. If the Prebytery approves, then the candidate(s) is ordained by the Presbytery.
Personally, I lean toward #1 since those who desire the Eldership are best known by those among whom the man associates with, worships with, etc. The #2 process removes much of the personal element of the local church which I believe is most important.
Of course, there is another case, #3 where a man is sent to a local church by a Presbytery who allegedly has examined/vetted the man and then the congregation expresses their like or dislike of that man to the Presbytery, etc. I've known of far too many cases where a local church suffers serious problems with men preaching heresy and recourse to have him removed/fired/challenged is very difficult.
Just my
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