The question of biology and shapeshifting is, as one might expect, largely a modern concern. More specifically, it tends to be a greater concern for urban fantasy and paranormal romances than for more “traditional” fantasy as the former two genres bring in more modern scientific views and foundations. Some, of course, dodge the question entirely, such as Jack Williamson, in Darker Than You Think (1948), who used lycanthropy as psychic projection—though it is unclear whether the body remains behind, is transformed, or something else, especially as the story progresses.
That said, the earliest exploration of the biology of shapeshifting that I’m aware of was produced by G. Havers in 1664. Havers published an English translation of A General Collection of Discourses of the Virtuosi of France, Upon Questions of all Sorts of Philosophy, and Other Natural Knowledge, Made in the Assembly of the Beaux Esprits at Paris, by the Most Ingenious Persons of that Nation (hell of a title). The “virtuosi of France”, according to Havers, argued, “[f]or otherwise, how should the Sorcerer reduce his Body into so small a volumn as the form of a Rat, Mouse, Toad, and other such Animal into which it sometimes is turn’d” (204). In other words, in the mid-17th century, they were arguing from a position that employed the law of conservation of mass (before said law had been codified).
Among others, Philip Jose Farmer built on this question in his short story “Wolf, Iron, and Moth” (The Ultimate Werewolf, ed. Byron Priess, 1991). He writes, “Only the moon saw his hair and skin melt until he looked like a mass of jelly that had been formed into the figure of a man [. . .] The furious metabolic fires in that jelly had already devoured some of the fat that Varglik had accumulated so swiftly” (59). Nina Kiriki Hoffman does something similar in her story “Unleashed” (The Ultimate Werewolf, ed. Byron Priess, 1991), “Change gripped her breasts, flattening them against her chest, her body shifting to absorb and redistribute tissue” (76). Obviously, both authors are concerned with the mass and tissue changes involved in changing from a human to wolf shape, and vice versa.
Farmer’s story also touches on the scientific question of energy requirements and use to change. He writes, “The furious metabolic fires in that jelly had already devoured some of the fat” (59). Charlaine Harris also plays with this briefly in her Southern Vampire series. Other approaches have included a strong urge to eat after changing shape, particularly repeatedly in a short span of time, as food and fat reserves are burned to fuel the transformation.
Some authors go a few steps further in linking biology and shapeshifting. For example, Ilona Andrews states that, at least for Lyc-V (Lycos virus) shifters, there are only mammalian shifters (Magic Bleeds). The implication is that because humans are mammals, they can only transform into mammals. Some exceptions are included later, but appear to be either a) non-human species (lamassu) or b) incredibly ancient or mis-identified (an apparent were-croc, which might not actually be a were/lyc-V case). Others have used this as well, including the webcomic Peter is the Wolf (it’s title a play on Peeter Stubbe, the infamous German werewolf, and “Peter and the Wolf”).
The last element that comes to mind for shapeshifting and biology is the actual reshaping of the body. Many authors choose to gloss over the change (ex. Pratchett) or gloss over it for some shifters (ex. Rowling for animagi). But, a few use the change for dramatic or horror effect. Charlaine Harris, for instance, writes, “It was a sort of gloppy sound. Sticky. Like stirring a stiff spoon through some thick liquid that had hard things in it, maybe peanuts or toffee bits. Or bone chips” (Dead to the World, Ace, 2011, p. 158). The painful bone reorientation is the key element here. Likewise, J.K. Rowling describes, in broad strokes, a similarly painful change as Remus Lupin is chained to Ron Weasley and Peter Pettigrew, emerging from under the Whomping Willow. The change is described as being highly painful previously as well, when Lupin describes his childhood transformations. These painful shifts are in contrast to the instant, silent, and painless transformations undertaken by the animagi. I suspect the difference is that in Rowling’s world lycanthropy is essentially a disease (although she switches back and forth between talking about it as an illness or a species), while animagi use a transfiguration spell.