When I am out on my long rides (on my road bike) my mountain bike shoes are clipped into my eggbeaters. My feet have been falling asleep after 10 miles or more. Is it possible that my clips aren't adjusted properly? Am I not spinning well enough? What's the problem?

Please advise.

This is old msg but if you still need advice on it, here goes.
I have similar prob, but I attribute it to shoes too tight combined with angle s etc. The combo seems to do it. I open my shoes and let my feet float more freely. This does the trick. However, it leaves me only loosely attached to the bike. Not terribly efficient so i use the method for temporary relief, then tighten up again.

Hope this helps.
JP

Your shoes are probably too tight (narrow). Loosen them up a bit or if that doesn't work you'll have to find a pair that is wider.

I agree with the shoes too tight as the main culprit, especially if they are new shoes. It may also be that the interior padding is wearing out in older shoes, especially if you walk on them a lot (as an example, running shoes are only good for about 100 miles before the inserts need to be replaced).

Alternatively, it may be that you are keeping the weight on the feet, rather than lifting on the upswing. This may be hampering the circulation. Make sure you are lifting on the upswing, instead of pushing up with the downforce of the other foot.
Note: I had that problem on an elliptical trainer at the gym (where the feet are always in contact with the pads).