You got me thinking, and I did some research and found a couple of "viable" routes between Everest's North and South base camps. From this poist: http://www.summitpost.org/phpBB3/wal...bc-t20583.html
You couldn't really 'walk' it around the peaks nearby. The closest you could get is to walk up the valley west of Gokyo, cross the Nangpa La, coming out near ChoOyu BC then walk across to Ev BC from there. Long, but quite possible. Tibetan refugees regularly do part of that.
Or if you could do some glacier travel, with maybe a little climbing, you go up the Gokyo, up the Nup La, down the West Rongbuk, back along the Central Rongbuk and to Ev BC. Shorter than the former option, but more technical. Hillary and Lowe did this illegal trip in '51 or '52, then Sayre and friends in '62. Don't know of anyone doing it since.
Expeds have climbed Khumbutse (west of the Lho La) to gain the Lho La, rather than climbing up to the Lho La from Ev BC Nepal side (v hard), so you could descend to north side Ev BC that way, but that is much harder and more technical. A Belgian climber did this illegally in the mid 80s - Nepal BC-LhoLa-Rongbuk Gl-Ev BC.
Note, you really want to read what happened at this pass's border in 2006:
After walking through "chest deep snow" a number of Tibetan refugees were fired on, some were killed, by Chinese border guards.
I dont think you could get your moto through, cool as it would be, and I dont think the Chinese will let you anyway.
Still, a cool idea.