How much truth is there to this statement or the spirit of it? He said it as a casual off-hand comment, it's not a history course.
Is there a historical basis for a spread of knowledge and ideas coming from the sack of Constantinople?
He was referencing the role of Greek scholars in the Renaissance. The Wikipedia page has some speculation in it, but it has a great references section.
I'm unaware of any examples literal caravans of scrolls, but I think it would fair to say that a lot of learned people, likely with whatever texts they could scrounge up, fled the Turks. They would contribute to the expansion of the arts and sciences that we generally term the Italian Renaissance. Many also fled to the self-styled Third Rome, Moscow. If there are any scholars who focus on that area, I'd love to learn more about it.