Members of the High Council have the responsibility to help strengthen and train leaders and members of the Elders quorum and High Priest group in the ward or branch in which they are assigned. Part of this responsibility lies in helping hasten the work of salvation by encouraging those who bear the priesthood to strengthen their brethren, particularly those who have left the gospel fold or become casual in their attendance and testimonies. The Church has a series of videos focused on how various church leaders play a role in hastening the work of salvation. Here is a brief video about the role that high councilors play.
Here’s the link to the video (I’d embed it but it kept auto-playing and until that is fixed, I’ll just link to the video).
The Church also has a short document describing the responsibilities high councilors over missionary work in a stake have regarding hastening the work of salvation. This is found here (as a PDF): http://www.lds.org/bc/content/shared/english/wwlt/hasten/hastening-the-work-high-councilor-eng.pdf?lang=eng