It's possible, and has been done before. It's even easier to handle with Wii games, as you could do it all over Bluetooth.
The problem with this, however, is synchronization. You're essentially running a copy of the software for each console and pretending that everyone is playing on the same console up to 4 times. So, after you account for latency(not only in the network, but in the I/O interface and its software), you still have to factor in that you have to beat 30-60FPS.
Not only that, but you have to account for AI. Because AI can be random, such as in the case of sports games, you'll be running up to 4 unique versions of a single game. So, while you might have scored a touchdown on one console, AI could've tackled you on another. The emulators get around this by hooking the memory of the host computer to make sure it's identical to the clients. It's more work, but you get accurate results.
Ultimately, though, it comes down to how much work you're willing to do for somewhat lackluster results. SSB was fun playing over Kaillera back when Melee was out, but you can get a similar experience now with Brawl.