Would it be fair to say this examination operates more on the principle of finding positive evidence to support their claim ("Define your faith and show it to me by what you do"), rather than an investigation looking for something to disqualify them?
In practice, this is going to depend on the leadership in your local church.
The objective would be several, I would think. Two or more elders confirming that there is a credible profession of faith in the person's life.
I take it you understand this is not an examination of being "good enough" to get in but rather, whether there is credible evidence, confirmed in the mouth of two or three witnesses that God has saved an unworthy person for His Honor and Glory. Also, one can't claim that and persist in an open, known lifestyle pattern of sin. (One of our PCA vows is to walk an orderly (obedient) Christian life- not a perfect one, but not an open one of defiance of God, either).
Implicitly, that means that the person understands and is relying on the gospel (not a mere notion that they are a Christian or a good person), that they are not in an open pattern of sin (such as "living together" out of wedlock), that they have or will be baptized (and that they understand the basis for that).
In the PCA at least, there is no doctrinal examination except what is necessary to establish a credible profession of faith (e.g. understands justification by faith in Christ's righteousness alone, even if they don't know that terminology). There may be a bit about baptism. A new member only vows to learn the church's doctrine peaceably.
There are some reformed denominations that require more examination of the church's confessed doctrine (but that is not the focus of your question as it relates to "examined for a credible profession of faith.")
If the person is not ready for membership in the visible church, they can be encouraged to further study, mentoring, etc. It's also a time to learn of their needs (which can be relayed back to the Deacons).
Remember, in the end, reformed theology takes a "high view" of the Church. While it may be common in broadly evangelical circles to think the church is merely a loose association of consenting adults, Scripture teaches a view of some authority in the visible church. Not authority to determine salvation, God alone controls that, but authority by which to govern and discipline.
It's an institution governed by officers whom He appoints, where we learn to get along, be accountable, and submit to the brethren. Sin will happen, and God will cause scandalous sin to become known. He gives the church some authority, not to determine their salvation, but to discipline it for the peace and purity of His Church, and for His Honor and Glory.
Oh, that God's people, by His grace would come to understand that they will one day be judged by their behavior. (Yes, even Christian will be "judged"- this life is not a game centered on our egos or thoughts).
One way it happened for us was two elders visited us at home. They asked us each to share our testimony. They asked a few questions about our baptism, church past, etc. Then, we had a prayer time- I think they prayed for us, and we prayed for them.
For us, it really was more a warm time of fellowship.
