A view held by some today is that because the church includes all who are in Christ in the whole world, it follows that elders are those who meet the qualifications set down in scripture, and they labor throughout the world as the Lord leads. Some would modify this to say that they labor in a city, since all of the churches mentioned in the New Testament appear to be confined to a city area (such as the church in Ephesus). While I agree that all who are in Christ will one day be together with the Lord, the limitations of time and distance do not allow us to dwell together with all believers at this time. My understanding is that the church is universal in scope but local in expression. This means that we certainly need to receive all who are in Christ as the Lord brings us into contact with one another, but the primary place we see the church expressed is in the locality in which we live. It is there that we can function as one body and be built together as a habitation of God. In my understanding, the locality is determined primarily by our ability to function together, not by man-made boundaries. God can make it very clear where He is gathering a people together to function as an expression of His body.
On the basis of this view of the church, elders are those who are raised up by God within a local body for the purpose of shepherding those in the locality. They know the sheep and the sheep know them. They know the details of the local situation and thus are able to make sound judgments pertaining to that particular body. They are intimately involved in the body in which they are recognized. They are "among the flock," not at a distance. They are not "super-spiritual" men scattered throughout the world to which we go for the final answer to our questions, but a plurality of brothers among us that we have recognized as mature spiritual brothers who can lead the flock.
It follows, then, that elders do not exercise their oversight outside of the locality in which they were recognized. They do not know the condition of other gatherings, and it is not their business to do so. It is not that they cannot travel to another place, but when in another gathering, they do not function as an elder. They may have another ministry (such as a teacher), but if they exercise that ministry, they do so at the invitation of that gathering and under the oversight of the local eldership. A brother who is visiting in such a situation will want to honor the local leadership in every way.
If a brother who is an elder moves to another locality for some reason and becomes a part of the assembly there, he would come in as a member and function as such, but he is not automatically an elder in that assembly. If, in time, the gathering chooses to recognize him as an elder in that gathering, that would be a choice for them to make in God, but he is not an elder in the new gathering just because he was an elder in a previous gathering. This leaves each gathering to function underneath the headship of Christ and to be responsible for the testimony in their locality.
A related question to this would be the movement of other members from one assembly to another. What part (if any) should elders have in the movement of members? I think if we consider the example of the natural family, it can help us. Suppose there are two families that live close to each other with several children in each family. Would we consider it a healthy situation if the children just stayed at the "family of their own choice" each night? I think we can agree this would not be good. Children can visit another family, but they belong to one family. I think this is a good way to think about the spiritual family. Believers may visit another assembly from time to time, but it is not healthy to "jump around." Our movement needs to be orderly, and with purpose and pure motives. In the case of the natural family, the movement would be under the supervision of the fathers of the families. In the spiritual family, the movement of members should also be under the care of the elders in each gathering. This is not to restrict fellowship, but to encourage all movement to be with purpose and unto edification for all involved.