I have to wonder what the point of that would be. It might catch a few extra wide sales but certainly wouldn't build a wide fan base. And I were committed to one of the other outlets and kept seeing an author popping up with a book available only for a limited time, I'd be annoyed and would pretty quickly start ignoring that author. I'm pretty sure I wouldn't be the only one. At the very least, it's hard to make a series work that way.
You know what I'd do, if I liked the author's books and didn't shop on Amazon? I'd buy the book before it went off sale.

I'm sure there's enough of those people to make it worth it for some (probably more, since it's pretty much been proven that scarcity works as a sales tactic even if it seems silly to some of us that it does). I mean, I hate it, and yet I am as susceptible to its pressure as the next person when something I really want is on the line.
I'm also sure that some people take the opposing perspective too, that's it's kind of good of the author to make sure people who don't like shopping at Amazon have the opportunity to pick up their book without forcing them to choose to forego it or support a retailer they don't like.