The Adoremus Bulletin has a brief article online about a response from the Congregation for Divine Worship and the Discipline of the Sacraments (CDWDS) concerning the common practice (in the US, at least) of giving blessings to people who come up during Communion but are not receiving Communion.
While the response (Prot. N. 930/08/L) says that "this matter is presently under the attentive study of the Congregation" and does not render a verdict, so to speak, the response does include several "observations", the first of which is that the "liturgical blessing of the Holy Mass is properly given to each and to all at the conclusion of the Mass, just a few moments subsequent to the distribution of Holy Communion."
It should be noted that the intention of giving blessings (especially to little children) is a good one; the placement of those blessings during the Communion procession is one of the issues, though. The Communion procession is for receiving Communion, and any blessing requested or received during that time is "out of place" and shouldn't be misconstrued as a "replacement" or "substitution" for Communion. Perhaps the traditional practice of staying in one's seat and making a "spiritual communion" could be promoted once more.
Another issue is whether extraordinary ministers of Holy Communion (laypeople) are permitted (or even capable) of bestowing such blessings in the liturgy. EMHCs are commissioned for the distribution of Holy Communion, which does not involve a blessing. The response from the CDWDS made the observation that lay people (EMHCs included) "within the context of Holy Mass, are unable to confer blessings" and that these blessings "are the competence of the priest".
(Side thought... The old formula for the distribution of Holy Communion was, I think, a blessing: "May the Body of Our Lord Jesus Christ preserve your soul unto life everlasting. Amen." With such a formula, which is still found in the Extraordinary Form of the Mass, I don't think a lay person could be a minister of Holy Communion.)