The transmission is well documented and worth following, because each stage added something identifiable.
Greco-Egyptian alchemy at Alexandria in the first three centuries CE is the earliest recognisable form, combining Egyptian metallurgical and dyeing craft with Greek matter-theory and a Hermetic religious framing. Zosimos of Panopolis, around 300 CE, is the first alchemist whose writings survive in quantity, and his work already contains both operational recipes and visionary allegory in the same texts, which is a useful corrective to anyone wanting a clean separation.
The Islamic world made it a science. Jabir ibn Hayyan, whose corpus is large and of disputed authorship, supplied the sulphur-mercury theory and a systematic experimental programme; al-Razi classified substances and apparatus with a precision that reads as recognisably chemical. The word alchemy itself carries the Arabic article, and so do alembic, alkali and elixir.
Europe received it by translation from Arabic in the twelfth century, and it became a learned pursuit with royal and ecclesiastical patrons and a steady problem of fraud, which drew prohibitions and, in England, a statute against multiplying gold in 1404. Paracelsus in the sixteenth century redirected it toward medicine, founding iatrochemistry and introducing mineral remedies against the Galenic orthodoxy.
Alchemy did not so much end as divide. Through the seventeenth century it was still respectable, Boyle and Newton both pursued transmutation, and the separation of a respectable chemistry from a disreputable alchemy was made in the eighteenth century largely by chemists who wanted to be taken seriously. What survived on the other side of that split was carried into the nineteenth and twentieth centuries by occultists, and then by Jung, and it is that line rather than the laboratory one that most modern practitioners inherit.