Many polytheistic religions have gods with their own powers, domains, etc. But I've found very few examples of myths of how one becomes "God of" something, especially for those that are born as gods and not deified.
Additionally, I know that Greek mythology changed over the centuries, and there are many versions of different myths. Did any gods "gain" domains? And if so, were they recognized as having newly gotten them, or were they only newly recognized for always having them?
The Greeks seem to have usually imagined that the gods' responsibilities were inherent in, or at least congenial to, their nature.
There are two stories, both attested early in Greek literature, that suggest the gods were assigned their responsibilities. One is the myth you mention about Zeus, Poseidon, and Hades casting lots for lordship (Iliad 15.186-93). The other is Hesiod's statement in the Theogony that Zeus gave the gods their responsibilities (881-5 ). Many other myths, however, suggest that the gods were naturally suited to their responsibilities - an idea that already appears in Homer, whose Zeus is inherently stronger, and thus better qualified to lead, than any other god (e.g. Iliad 8.19f).
The birth of Athena is probably the most famous illustration of the idea that a god's responsibilities are intrinsic. Athena, infamously, was born after Zeus swallowed his first wife Metis (whose name means "cunning"). As an immortal, Metis could not die - she lived on inside Zeus - and being a personification of wisdom, she naturally gave birth (inside Zeus) to an exceedingly intelligent daughter. This daughter, Athena, emerged from the head of Zeus (helpfully cracked open by Hephaestus) both fully armored and fully prepared to assume her role as goddess of strategy and craft. Even less subtly, Aphrodite, goddess of love and generation, was literally born (at least in Hesiod's account) from the severed genitals and semen of the sky god Uranus.
Some gods were aware from birth of their responsibilities. In the Homeric Hymn to Apollo, for example, the god claims his domain with his very first words: "The lyre and the curved bow shall ever be dear to me, and I will declare to men the unfailing will of Zeus" (131-2). Hermes was equally precocious: "born with the dawning, at mid-day he played on the lyre, and in the evening he stole the cattle of far-shooting Apollo" (Homeric Hymn to Hermes, 17-19).
To turn to the second part of your question - the responsibilities associated with various Greek gods did indeed change with time, though these shifts tend to be nearly invisible in the myths. Although the Mycenaeans worshiped many of the familiar Greek gods, they associated many of them with rather different domains. It seems likely, for example, that Poseidon was originally an earth god (preserved in later myth by his association with earthquakes).
The Dark Ages and early Archaic period (an era of considerable Near Eastern influence) were characterized by extensive religious innovation. But once the poems of Homer and Hesiod became established as the closest thing the Greeks ever developed to theology, the literary image of the gods was more or less fixed. Locally, however, individual gods continued to be associated with a much wider range of responsibilities and prerogatives than those mentioned in the most familiar myths.
Edit: added reference to Hesiod suggested by u/Linred
