He is a reference to what Google says it's supposed to be used for as well.
http://googlewebmastercentral.blogspot.co.uk/2009/02/specify-your-canonical.html
I personally would say it would have the same effect as if you did a redirect on Domain B pages.
Domain B would rank but Domain A would drop off.
I've never tried it but I'm taking an educated guess.
If Domain A was sitting on say a exact match domain and Domain B was not, then I don't think rankings would be mirrored. So it's not a straight forward flip.