I have heard that K31s have been succesfully converted to .30-30. Is anyone familiar with these conversions? How is it achieved. Are there any magazine modifications needed?

That would be tough, very tough! Check out the 7.5x55 compared to the 30-30. I don't think so.
http://www.chuckhawks.com/7-5x55_Swiss.htm
http://en.wikipedia.org/wiki/.30-30_Winchester

I'm going to delate your other post tango. The 7.5 by 55 is physically bigger than the 30-30 case. You'd have to rebarrel I think.

Even if rebarreled you may also have to open the bolt face from 0.496 to 0.502 to fit the rim.

??? Why turn a rifle with more power than a 308 Win into a 30-30???

If $$.$$, why not just reload the 7.5x55? A tumbler, press, dies and cases would cost less than a new barrel!

I may be a touch off target in saying it was a K31. There is a Schmidt-Rubin 1911 sporter in .30-30 for sale here: http://www.accuracyplus.biz/usedguns.php?sort=2 at Item # 0967 (no pic). I'm almost certain I saw a .30-30 K31 selling on GunBroker recently....(?).

It's not something I want to do but it did grab my attention.

There were commercial conversions of the 1911. I had a few in my shop back in the 80's. These were converted by sleeving the chamber end of the 7.5x55 chamber with a 30-30 length sleeve. This left an ungodly leap for the bullet between the sleeve and the original throat. Avoid these conversions. ~Andy