While evangelists were given the qualifications for selecting the shepherds, the implication is that they stayed as long as necessary to establish the foundation, and appoint the elders before going on to the next city. Travel was one of the advancements of the Roman empire and people moved from city to city. The list of gifts in Ephesians 4:11-12 were given in the order in which they are listed for the same basic purpose and duration.
It is not clear that the elders were restricted to living only in the city in which they lived when they were selected. They may have moved as Pricilla and Aquila moved. The selection only implies the intent to have elders in every city of disciples, especially where there were few spiritual gifts present. The on-going work to teach and prepare brethren for service rested upon these men just as the other gifts mentioned in Ephesians 4 (apostles, prophets, and evangelists).
One argument in favor of the view that these men were not as mobile is seen by the fact that these were family men, according to the domestic qualifications given by both Timothy and Titus. They also had been in the city long enough to acquire the good reputation of being “above reproach”. However, in response, we could point out that Philip, the evangelist, had a family in Caesarea. He’s found early in Luke’s’ writing where he moves to Caesarea and is still there toward the end of Luke’s record. It is only the opinion of this writer that, seeing they bear the same role and have the same purpose and duration (Ephesians 4:11), at least one of the gifts (either apostle, prophet, or evangelist) would live in a city with disciples to equip them for service and edify them.