Outline of the concept
The idea of ablations is introduced in the third volume of Return to Labyrinth, and is explained by the sorceress Mizumi who is the only character with the ability to create them. Mizumi tells Toby and Moppet that the creation of an ablation involves the extraction of a particular essence from a chosen subject, the essence generally being a powerful emotion or characteristic. The ablation is solely charecterized by the stolen aspect it embodies, while the source it was taken from loses that aspect.
The ablation and its source are linked, enabling them to reach one another if they so will it but also meaning that if one dies the other dies as well.
It is possible for the ablation to be re-absorbed by its source, making the source whole again.
Role in Return to Labyrinth
Mizumi is the first character in the series known to have created ablations, extracting the feelings of Hope and Regret from her to make her daughter Moulin and Drumlin. Both first appear as infants, and are later seen as children implying that Mizumi created ablations that age as real children would. Mizumi made these ablations following Jareth's rejection of her; she was unable to cope with her emotions and removed them so she would no longer have to experience then. Mizumi eventually re-absorbs both Moulin and Drumlin, making herself complete again. As she is once again able to feel hope and regret she withdraws her hostile from the Goblin Kingdom, resigning herself to wait in the hope that Jareth will one day return her love.
Jareth travels to Mizumi's Kingdom of Moraine when Moulin and Drumlin both appear to be children, asking her to make an ablation of Sarah's dreams that he can take back to his Kingdom with him. Mizumi creates Moppet, giving her to Jareth in exchange for a wager that he will give her his Kingdom and its King if he fails to make Moppet love him. Moppet was formed from Sarah's dreams, however Sarah's dreams were crushed when she received a rejection letter from Julliard school. Consequently, Moppet mostly appears melancholy and unhappy. She is unresponsive when Jareth begs her to love him, and after many years spent trying to win her love he abandons her in the Junk City so he can devote his energies to pursuing the real Sarah. Moppet once again becomes a part of Sarah towards the end of the final volume of the series, and expresses happiness at the prospect of finally experiencing a sense of wholeness.
Whereas Moulin and Drumlin are shown to age from infancy through to young adulthood, Moppet was given the appearance of a teenager upon her creation and does not age any further. This implies that the growth rates of ablations is something that can be set.