We have been waiting for a while to see the on-prem story. It was actually taken off the road map for a while and today it was finally disclosed. It was by no means what we expected an on-prem solution would look like… There will not be an “offline” stand alone on-prem installation running in a local data center by it self. It was so much better!
The new on-premise solution will focus on having individual business processes (Workcenters) available that can live on their own in offline situations, but moving data back up to the Azure hosted solution when available. Giving the solution architecture the possibility of having multiple on-prem installations of different processes on the same solution in many locations.
The offline Retail solution that was disclosed at Ignite 2016 will be available with the launch of Dynamics 365 in November, while the other processes are rolled out during 2017, starting with Shop Floor Control and warehouse management.
So you can have an on-prem warehouse solution in LA, Washington and Tampa on site at the warehouses, An on-site project management solution on the off shore platform in the Mexico Gulf, 25 retail store locations across the nation and an on-prem manufacturing solution for your factories in USA, China and Germany … Choose and pick what you need!
On-prem solutions will be self contained offline applications running AX with databases and all the required business logic, but require a pipe to the Azure environment for other data and off cause all the Power BI, Analytics, Management reporter and cloud other stuff. The user experience will be indistinguishable and smooth no matter the status of the locations current internett access.
Watch this video with Christian Pettersen and Sri Srinivasan from 29:30 or get the entire intro with Christian Pettersen from 09:30.
http://www.ustream.tv/embed/recorded/92170675?html5ui
dyntips.com by Fredrik Sætre 
Great summary! and fast. Thanks for the update