Each time they change the price, they put a sticker with the new price and barcode over the old one, on every discounted item. The employee time required to do this is non-trivial. I imagine that in some places, stores would decide that it's not worth the effort, since a discount might not increase sales enough to match the reduced profit per item (especially in affluent areas where customers can easily afford the full-price item, and maybe turn their noses up at items near their expiration date).
At a grocery shop where I usually purchase groceries, on the expiry day, some products will have a 15% (or so) discount.
I don't remember seeing this in other shops though, or rarely.
I think they can't legally sell it if it's past the due date though.
It's a common technique to pay staff to stock shelves in such a way that the goods about to expire are at the front, savvy customers will dig at the back to get the new stock. Right now there's no price incentive not to do that.