In Civ IV that would lead to overlapping quotes, it made a ton of noise and was really annoying... So, for this to be back, they'd have to make it so that the quote gets interrupted if another quote appears.
In other words, if you just reached the medieval era, then you get two quotes: the tech you just researched's, and the era's. Currently, closing the pop-ups shuts the quotes, even if nothing else would get in the way. A-la Civ IV, closing the pop-ups would not shut up any quote, making the era quote start while the tech quote is still going, making a huge mess on the sound. The way I think is best, closing the first pop-up would silence the first quote because the second pop-up also has a quote for it to say, but closing the second one would not because no other quote would overlap it.
On a similar note, I'd really wish leader's themes would stay after closing the leaderscreen. These themes are absolutely beautiful, but when diplomacy can be made in a question of seconds, we might not get to hear much of them during play.
There's already a bug that makes just this: sometimes (not sure what triggers it) the music from the diplomacy continues on even after closing, and I'm always happy when it does that because then I can hear to their theme while playing.

Like with the quotes, the music would silence and give way to another if I made diplomacy with another leader.
Finally, make our own leader's peace music be one of the selectable ones while in peacetime. Currently, our war theme can randomly play while we are at war, so why not our peace theme while we are at peace?