I can’t remember whether it has been addressed all that much, Invernessian, but it probably has. I’m getting old… 
I would say that in most cases in the NT, whether the gospels or the epistles (particularly of Romans), the elect refers primarily to the Jews, secondarily the church in this world. As it seems obvious that not everyone receives salvation in this life, there may possibly be something for the elect to do in the age(s) to come in shepherding our younger brothers and sisters home to the dear Father who waits for them with with love.
Maybe we’ll get to do that, maybe not. I hope so–I’ve never been much good at evangelism in the sense of proselytizing the (at least apparently) unsaved during this life. I’ve always wanted to help, but the nonchristians around me haven’t typically been receptive to that kind of help. I’m most likely inept or too timid to push, I suppose. At any rate, I’d like a chance at it with better knowledge and wisdom and effectiveness than I have now.
But… back to your question… I think the elect concept is kind of like nesting dolls. You know, the traditional Russian dolls that open up to reveal a smaller doll, and another and another…
The elect are a select set within the world–say Israel. And within Israel, the Levites, and witin the Levites, the sons of Aaron, and within the priestly family, the high priest–and then God says, "I’m going to give you another priest after the order of Melchizedek. Jesus is of the people of Israel, but of the TRIBE of Judah (not Levi). He is a Jew and the ultimate representative of the nation of Israel, but ALSO of the race of Adam (the chosen seed). He came through Judah, but he is high priest after the order of Melchizedek (a non-Jewish person who was nevertheless venerated and tithed to by Abraham before the founding of the Jewish nation/race/religion/law). Lots of people (including me) think that Melchizedek was likely a Christophany, of course, but no one I’ve heard of thinks he was a child of Israel–at least not until He (if he was a Christophany) was born of Mary.
I’m getting off-track again. 
This is where I was going anyway. If you close down on the elect, the ultimate election belongs to Jesus. He is the representative not only of the Jews, but also as the seed of the woman, the representative of the entire human race, and as being “after the order of Melchizedek,” high priest to the entire human race. So . . . the church is the next stage (or maybe the other side) of this nesting doll in that we are “in Christ” and are called to the ministry of reconciliation to the world.
The nesting doll has gone from one woman (Eve) to one couple (Eve & Adam) to, essentially, the whole population of human beings. From there, the population contains the nation of Israel, which contains Mary, through whom comes the God/man, the Christ. Christ then expands the elect to include all those “In Christ,” and then all those to whom this priesthood is called to minister and ultimately bring into the sheepfold. It reminds me of “The Last Battle” (the concluding book to CS Lewis’ Chronicles of Narnia) in which he paints the picture that the further into Narnia you go, the vaster the country becomes. “In my world,” one of the characters says, “a tiny stable once contained something larger than our whole world.”