Where can I find a teaching on the reason extraordinary ministers of holy Communion cannot give blessings to those who approach the altar for a blessing at Communion? We are trying to teach our EMHCs to stop this and would like to have a resource to help.
So far as I know, there is no “teaching” because the whole practice of distributing blessings to people who approach for a blessing during Communion is not supposed to be occurring in the first place. There is no provision for it in the rubrics of the Roman rite. Some parishes do this, but it is strictly extra-liturgical and, when done, should only be done by the priest. (Even a deacon serving at a Mass should not do this.)
If the pastor really wants to put a stop to this in his parish, he is going to have to re-educate the congregation, including making a specific announcement before Communion is distributed that those adults who are not receiving Communion for whatever reason should remain in the pews. (Children too young to remain in the pews may walk with their parents without receiving Communion or a blessing.)
As for the extraordinary ministers of holy Communion, they serve at the discretion of the pastor. Ask the pastor to instruct the EMHCs not to bless people in the Communion line and inform them that if they do then they will no longer be delegated to distribute Communion during Mass. If the pastor wants to continue to dole out blessings during Communion, then the EMHCs could be permitted to simply say to those seeking blessings, “May God bless you,” without touching the person or making any kind of hand gestures.