I use Firefox to log into my Stack Exchange account(s) and the login handover between Exchanges works well - for vanilla Firefox.
A while back, I installed the Self-Destructing Cookies add-on for Firefox. (An absolutely superb plugin, I highly recommended it).
For sites that pass credential cookies between affiliated domain/sites (and where I want this to happen), I usually just add the applicable domains to the plugin's whitelist, after examining the Live HTTP Headers log.
However I can't seem to cover the whole chain involved in an inter-Exchange login handover. I currently have
openid.stackexchange.com and my topic specific exchange domains white-listed. But I get asked for login credentials. Oddly enough if I back-navigate then forward-navigate from that screen I'll be logged in from cookies correctly.
This behaviour can be pretty easy reproduced with a Firefox browser with Self-Destructing Cookies add-on installed (and any cookie manager add-on for easier cache control).