The authority in any gathering of the true church is first heavenly, and then expressed locally. Christ is the head, which is heavenly authority. Elders express His oversight, and must give an account to the head for souls in the local gathering.
When we speak of elders, too often we make them something they are not. They are not the head. They have never been given a place to dominate or rule the body of Christ. Their ministry is one of caring for the body, watching over, feeding, leading by example. Their authority is purely spiritual; that authority is determined by the grace God gives as the brother moves in the revelation God has given him. Elders are brothers amongst brethren with grace to function under the direction of the Holy Spirit. Each one in the church needs to "recognize those who labor amongst you, and who are over you in the Lord, and admonish you; and to esteem them very highly in love for their work's sake" (1 Thessalonians 5:12-13). But these brethren can never take the place of Christ or the Holy Spirit. No man or program can take the place of the Spirit.
Eldership is an office which is local in nature and must remain local if God's purpose is going to be fulfilled. When any man or group of men creates an authority which is extra-local, they interfere with the testimony and witness of the local church. They also move into denominationalism, which destroys the testimony that "Christ is one." Any authority that man creates removes the gathering from spiritual and scriptural grounds of expression.
All workers and ministries first come under the local oversight of elders. Otherwise, if a ministry is under the authority of one outside the gathering such as an apostle or teacher, it destroys the expression of locality and gives control to man rather than Christ. For the local expression, Christ's authority works through local elders, not through extra-local brothers. Any spiritual authority working through men that have been called by God is always for the purpose of building up the local expression, never for controlling one or more local churches.