Great news: Astro officially confirmed that the whole Astro range will be compatible with the next gen consoles!
Both sound (dolby) and voice will work on both platforms. For PS4 you'll have to install the post-launch PS4 firmware update (your PS4 will do this automatically) and for the Xbox One you hvae to buy the official headset adapter that will be made available by Microsoft at launch.
Quickly the details:
PS4: We can say with 100% certainty that our headsets will deliver Dolby ® Digital GAME AUDIO on the PS4, just like they do for the PS3. Our headsets will also deliver full voice chat functionality over the PlayStation Network upon Sony’s release of a post-launch PS4 firmware update.
Xbox One: Game Audio for the Xbox One will function as normal, through the optical output – meaning all our current products will deliver full Dolby ® Digital GAME AUDIO on the XBOX One. However, the chat connector on the controller is now a proprietary connector, meaning that our current products will require an adapter to connect to the controller for Xbox Live voice chat. Fortunately, until this chat adapter is available, the built-in Kinect microphone will provide full voice communication access to Xbox Live while using ASTRO headsets on the Xbox One.