If we read II Corinthians 10, Paul and his coworkers had a measure of rule that extended to Corinth because they had gone as far as Corinth with the gospel of Christ. They wanted to extended their measure of rule and they wanted the Corinthians to help them do so by helping them reach further unreached areas.
This was in contrast to others who called themselves apostles who boasted about the Corinthians, but the Corinthians were not the fruit of their labor.
One aspect of NAR teaching, is that some of them, at least a decade or two ago when this was becoming a thing, were going around trying to talk pastors into coming under their wing. The idea was to put another layer of hierarchy on top of existing church structures.
Let's think about a lot of the existing evangelical church structures. In the Bible, the apostles appointed more than one elders from within the church/city they would minister. Paul described them as bishops/overseers. They were told to pastor the church/flock of God. Elders had to meet a number of lifestyle and spiritual maturity requirements. are to be 'apt to teach' in one set of requirements. The ones who rule well are worthy of double honor especially those who minister in preaching and teaching.
The Bible does not assign one elders to give a long talk every meeting. I Corinthians 14:26 says, 'Every one of you has a psalm, has a doctrine, has a tongue, has a revelation, has an interpretation.' And what was Paul's commentary on that- 'Let all things be done unto edifying' followed by some other specific instructions.
Many modern evangelical churches already have highfaultin titles that aren't in the Bible like 'Senior Pastor', when Peter addresses elders and tells them to pastor, he calls the Lord the chief Pastor.
One of the NAR teachings is that there were certain men gifted by God as apostles, and that church leaders are supposed to come under their influence and let them pastor the pastors. The problem is, where does the Bible teach this? If someone is gifted as an apostle, let him go preach the Gospel. There are missionaries and church planters who believe God has called them, sent them, directed them (or whatever they call it) to do their work, and often it is confirmed through other saints, who go out and start new churches. Some take the Gospel to unreached groups of people and areas. It makes sense that they have a 'measure of rule,' help appoint elders, etc. Isn't that the sort of thing that Paul and his coworkers did?
The Twelve apostles helped get things going in Jerusalem, but stepped back from it. Eventually elders were handling donations and the Seven were feeding the widows. They let James and the elders handle the political situation with Paul. The Twelve ministered the word, prayed, and were witnesses of the resurrection. Paul also was very much involved in the formation of churches. Churches formed and were churches without elders. (That's possible if you have meetings like those described in I Corinthians 14:26 and Hebrews 10:24-25.) Then he and/or his coworkers appointed men from within the congregation to serve as elders.
Toward the end of Acts 14, after Paul and Barnabas had finished up their journey by going back and appointing elders in some of the churches, it says that had fulfilled the work. In chapter 13, we read that the Spirit had called them to the work. So if they had fulfilled it, we might take that as an implicit approval of how they handled the elder situation-- letting the churches exist first, then appointed elders from within the very congregations they would pastor, more than one of them. Paul later told the elders of the church in Ephesus that the Holy Ghost had made them overseers (bishops.)
I don't see a Biblical case for the senior pastor system and the emphasis on pulpit, pew, teaching being packaged into one-man 45 minute sermons. But does that mean that God's grace does not work in the lives of people in these churches to save their souls, to minister in spiritual gifts, or to bless others? I believe it does. And while I disagree with some of the NAR teachings about adding another layer of hierarchy to this questionable system and calling these church-network heads 'apostles', do I have any reason to think that God will not work in their lives to save their souls, minister in spiritual gifts, or bless others? No.
I also realize this is a wide and diverse movement, and a church might get some influence from it without NAR being a part of it's identity, so it is a complicated issue. I also do not know if there are still people going around trying to recruit for the NAR, or if that was a phase a few years back and they got their networks now.
There are also apostolic networks that think of apostles as 'mover and shaker' type Christians in different realms of society, business, education, etc.
Ephesians 4:11 is in the Bible. As Christians, we should accept that. But it does not seem much effort has been paid to actually studying out what an apostle is in scripture.
the understanding that Jesus did not only call people publicly but he called to do and learn of Him also. Jesus has to call you, save you before He can disciple you 
