The Evangelical Free Church does not permit "elder-rule" governance in our churches, and I hope we never do.
The "elder-rule" movement seems to have started at Dallas Theological Seminary during the late 1970s, but has picked up a quasi-Reformed coloration to it over the decades. The essence of it is that the elders of a church are self-perpetuating and self-disciplining. Elders in a church self-appoint, and they cannot be removed by the congregation. One well-known church group that follows it is the Sovereign Grace churches. A second group that uses this leadership system are the Fellowship Bible churches, which were fostered by Dr. Gene Getz of DTS.
The "elder-rule" method, by insulating elders from congregational power, does not follow Scripture. Even under the Old Covenant, which was heavy with kings and priests, the people had the authority to execute judgment on their own leaders. To take life is the ultimate exercise of earthly authority. By implication of Moses' law, an Israelite elder who tried to turn the nation away from the living God would be executed by the people, not just by other elders (Dt.13:10). The people executed anyone who worshiped Molech (Lev. 20:2). God said they were all a royal priesthood (Exodus 19:6), not just the elders. The people of Israel were the congregation of the Old Covenant, and the people had the God-given right and responsibility to execute their own leaders under certain specific circumstances!
Christ continued this rule into the New Testament era, though He changed the implementation of discipline from execution to excommunication. God gave Israel the power of the sword, but He did not do so with the Church. Jesus was clear, in Matthew 18:15-20, that God assigned the power to evaluate and excommunicate to the entire congregation, not just with the elders. Paul reaffirms this rule, with a caution, in 1st Timothy 5:19-20. On one hand, Paul forbids the accusing of an elder on light or insufficient testimony (5:19). On the other hand, those elders who are found to be sinning should be publicly rebuked (5:20), and probably are removed due to no longer being above reproach (which is the standard in 1st Timothy 3:2).
This is why the "elder-rule" system lends itself to cronyism and abuse -- it is not accountable to the congregation. It also promotes church splits, since a congregation protesting elder-incompetence, malfeasance, or doctrinal error, has no alternative but to knuckle-under or leave. (A claim that an elder board will always take care of such things themselves is naive).
God is wise. He set up a system of checks-and-balances for the Church, which we ignore at our peril. A rebellious congregation is a shame on the name of Christ, but so is a system of lords-n-peasants. People should not be tenant-farmers in their own church! This is one reason why liberalism spreads so quickly and so decisively in episcopal and presbyterial systems -- as the episcopal fish rot from the heads down, the rot spreads because the bodies have no administrative authority to stop it.