Sure, the Unreal Engine 3 demo (Epic Citadel) had been already ported to Flash one year ago. http://www.unrealengine.com/flash/ Since ActionScript VM is far slower than JavaScript, I'm afraid I have to say the Mozilla's demo does not prove the performance of asm.js at all.

And here's the latest Unreal Engine 4 demo. http://www.youtube.com/watch?v=dO2rM-l-vdQ

I know there are improvements on the web platform each year but I wish people would be very very cautious to say "we've achieved nearly native performance!".

To begin with, I think it's difficult to run 3D games (including Flash and Unity) on browsers because of the asset loading (and AOT compilation) time. If people must wait for seconds and even minutes to start a game, we can't do business with it. As many web developers says, immediate page loading is must. I'm personally running Flash and HTML5 games and I observe 80% people leave just in a few seconds loading.

To solve this issue, game stores (PS3, Xbox 360, App Store, Google Play, Steam and many others) are using the "reserve download and play it later" model. After all, traditional install apps are not that bad architecture...