The open beta for Battlefield 4 will begin on October 1st. It will be available on all current-gen platforms: PS3, Xbox 360, and PC. Next-gen doesn't get a beta because the consoles aren't out by then. Sorry folks!
It's unclear when the exclusive beta access included with Medal of Honor Warfighter will be made available. However, VideoGamer points out that BF3's exclusive beta opened 48 hours before it was free for all. History may repeat itself here.
And while the next-gen version won't receive a beta, EA wants to remind players that any stats earned in the current-gen version will transfer into the next. And Battlefield 4 will be available on day-one of both console launches. And like so many cross-gen games, $10 will upgrade you from a current-gen to next-gen version.