Unless I'm mistaken I don't think that's really possible. Most cookies, for security reasons, can only be shared across the domain (you can go across subdomains, such as www.example.com and store.example.com) but actually changing domains would be a problem.

One possible solution would be in all of your sites' settings.php files, set the $cookie_domain URL to one single domain that you would use as your "base" site. That might solve your issue, but you'd have to test first.