In our society, has there been a transition from older literary works to "culturally relevant" ones?
This point is certainly true. There has been a shift away from a standard western canon of literature to more culturally relevant one. This is part of the push of postmodernism. Foucault says this brilliantly in an article where he argues for the importance of the archaeology of knowledge. With the idea of a dominant meta-narrative gone, people are now reading literature that they once would not have. Part of the reason for this is because no one has absolute knowledge.
All of this also fits in very well with a globalized world. Places like America are international places, especially large cities. Universities are now reflecting this as well. At one point the typical university curriculum was western. Now people can major in all sorts of non-western subjects and languages. Simultaneously we see the decline in western subjects, like classics.
All of this is say that the literature that we read is now more culturally aware and the voice that we hear are more global in perspective.
I'm not sure where I heard this, and I only paraphrase the expresssion, but it suggests the "All literature is true in some way for all ages and all people."
I have also taken it to mean that any literature, regardless of it's age, offers some morsal of wisdom (we call these themes) for any reader at any time in any place in the world. Is it true? That's fro readers to decide.
When I read your question of "cultural relevence" I suppose it to mean can old, fuddy-duddy, dead guy books or poems speak to young, hip, internet-surfing folks today? I say yes.
My proof. . . movies: Remakes of just about every Shakespeare play (both comedy and tragedy)--you probably don't even know you're watching the old bard.
They put Angelina Jolie in Beowulf, and they have a script for Huxley's Brave New World starring (maybe) Leo Dicapprio.
It's in books too. . .Pride and Prejudice (and Zombies)--and it's a series!
At the end of the day, good literature will speak to readers. You just have to listen to apply the themes to your culture in a relevant way.
Another unattributted, paraphrased expression I use comes to mind:
"There are no new stories, just old stories being re-told in a new way."