Tolkien wrote an essay on this subject in 1960, called "Dangweth Pengolod." The essay is rather long, so I'm not going to quote the whole thing, but the highlight for me is this passage:
[T]o the changefulness of Ea, to weariness of the unchanged, to the renewing of the union: to these three, which are one, the Eldar also are subject in their degree. In this, however, they differ from Men, that they are ever more aware of the words that they speak.
A man may indeed change his spoon or his cup at his will, and need ask none to advise him or to follow his choice. It is other indeed with words or the modes and devices of speech. Let him bethink him of a new word, be it to his heart howsoever fresh and fair, it will avail him little in converse, until other men are of like mind or will receive his invention. But among the Eldar there are many quick ears and subtle minds to hear and appraise such inventions, and though many be the patterns and devices so made that prove in the end only pleasing to a few, or to one alone, many others are welcomed and pass swiftly from mouth to mouth, with laughter or delight or with solemn thought - as maybe a new jest or new-found saying of wisdom will pass among men of brighter wit. For to the Eldar the making of speech is the oldest of the arts and the most beloved.
History of Middle-earth XII The Peoples of Middle-earth PArt 3: "The Teachings of Pengolod" Chapter 14: "Dangweth Pengolod"
Thus, the Elvish languages evolve similarly to how mortal languages do, but in a more deliberate, guided way.
There's also evidence that Quenya borrowed from Valarin, the language of the Valar and Maiar:
The Eldar took few words from the Valar, for they were rich in words and ready in invention at need.
History of Middle-earth XI The War of the Jewels Part 4: "Quendi and Eldar" Appendix D: "*Kwen, Quenya, and the Elvish (especially Noldorin) words for 'Language'"
What effect that would have had on Sindarin is unknown, especially since Thingol banned its use, but it is evidence that the Elves were willing to borrow from other languages, when it suited them.
There's further discussion on this subject in The Peoples of Middle-earth, where Tolkien remarks that the Elvish tongues in Middle-earth did indeed diverge from those in the Undying Lands:
All things changed in Arda, even in the Blessed Realm of the Valar; but there the change was so slow that it could not be observed (save maybe by the Valar) in great ages of time. The change in the language of the Eldar would thus have been halted in Valinor; but in their early days the Eldar continued to enlarge and refine their language, and to change it, even in structure and sounds. Such change, however, to remain uniform required that the speakers should remain in communication. Thus it came about that the languages of the Eldar that remained in Middle-earth diverged from the language of the High Eldar of Valinor so greatly that neither could be understood by speakers of the other; for they had
been separated for a great age of time, during which even the Sindarin, the best preserved of those in Middle-earth, had been subject to the heedless changes of passing years, changes which the Teleri were far less concerned to restrain or to direct by design than the Noldor.
History of Middle-earth XII The Peoples of Middle-earth Chapter X: "Of Dwarves and men" Relations of the Longbeard Dwarves and Men