After the Peace, the Fraction Rite, accompanied by the Agnus Dei:
83. The Priest breaks the Eucharistic Bread, with the assistance, if the case requires, of the Deacon or a concelebrant. The gesture of breaking bread done by Christ at the Last Supper, which in apostolic times gave the entire Eucharistic Action its name, signifies that the many faithful are made one body (1 Cor 10:17) by receiving Communion from the one Bread of Life, which is Christ, who for the salvation of the world died and rose again. The fraction or breaking of bread is begun after the sign of peace and is carried out with proper reverence, and should not be unnecessarily prolonged or accorded exaggerated importance. This rite is reserved to the Priest and the Deacon.
The Priest breaks the Bread and puts a piece of the host into the chalice to signify the unity of the Body and Blood of the Lord in the work of salvation, namely, of the Body of Jesus Christ, living and glorious. The supplication Agnus Dei (Lamb of God) is usually sung by the choir or cantor with the congregation replying; or at least recited aloud. This invocation accompanies the fraction of the bread and, for this reason, may be repeated as many times as necessary until the rite has been completed. The final time it concludes with the words grant us peace.
The distinction for what to title this rite is important.
The current piece of dispute is how to repeat and lengthen the threefold invocation to accompany the liturgical action. It can be done three ways:
Simply repeat the words as given until the action ends.
Treat it as a true litany by varying the address to Christ (Hope for all, Prince of Peace, etc.) and give the assembly the cue for the final response by returning to “Lamb of God.” This is the easiest and most pastoral approach, though I can envision two or three ways to accomplish a good cue with option one.
Vary the part, “you take away the sins of the world,” with other petitions. The most successful of these attempts in my experience was Donald Reagan’s Mercy, Mercy: Mass in a Jazz Style. The drawback to that setting is that the people sang the minimum, “have mercy on us/grant us peace.”
Have at it, friends.
Catholic Sensibility 
Well, putting aside the issue of licitness for the moment, if you want actual tropes, it would seem that, back when the Agnus was troped many centuries ago in other contexts, it would not be the invocation that would change, but the “qui” clause – that is:
Agnus Dei, qui tollis peccata mundi, miserere nobis
Agnus Dei, qui [X], miserere nobis
Agnus Dei, qui tollis peccata mundi, dona nobis pacem.
The idea of changing the invocation is what is an innovation, it would appear.
That said, I’ve not found it necessary to trope. All that is necessary is to provide a pregnant pause over a significant cadential cue for the final verse.
Since the shift in preparing the cups at preparation of gifts, I’ve not found it necessary either. With our new Mass of Saint Ann, we extend the introduction to six bars to state the whole tune, and the threefold repetition suffices. The hardest thing is to detach cantors too-well trained in Christ-invocations and battle their instinct to intone, “Lamb, X, Lamb.” Which strikes me as needless.