@Kit-Bishop @Russ-Terrell I don't have an Arduino dock but saw your post....
The Yun (and some other dual boards) communicate similar to what you would have to do with the omega. On the Yun, the linux part is not meant to be powerful and just really a gateway to the internet, but more can be run there. The communication is over a library called Bridge. This is also similar to if you purchased an Arduino wifi shield based on a ESP8266 or had one not in a shield. There are GPIO pins that are there, but not connected directly.
I am not sure how the GPIO pins would get connected directly. Are you wanting an input pin on the Arduino to register as an input pin on the Omega? In that case I suppose a board that didn't have an Arduino but just accepted the shields would accomplish that, but you would have all kinds of software issues then. The Arduino and the Omega are quite different in terms of timers, interrupts, multi-talking, etc.
It is an interesting issue that likely needs a hardware/software bridge - the dock being the hardware. I would be nice to have something where any Arduino shield could be plugged in and just work. Just not sure how that would work. You need to use specific libraries on a shield to work with it. I can see someone writing a I2C wrapper around a library for the Arduino so that the communication to the Omega is easier, but that would be on a per shield/library basis.
Who knows, maybe I don't understand well enough and the Onion folks have some cool ideas that would simplify the programming with Arduino shields.
PS. Several times I referred to Arduino, the Board, and Arduino when I should have just said Atmel Atmega AVR.