MacBook – move the logic board between different models of MacBook Air

I think you're out of luck. From iFixit, here's the logic board from the original MacBook Air:

And here's the logic board from a 2012 MacBook Air:

Unlike desktops, which typically use standardized parts, laptops use fairly specialized components, and on top of that, Apple tends to iterate its designs fairly aggressively even within models with similar outwards appearance to achieve better cost or performance (e.g. smaller logic boards means more space for the battery). Because of this, most Apple components beyond hard drives and RAM are only interchangeable with those of the exact same generation.

There are 4 different CPU that fit that model A1369 according to iFixit:

• http://www.ifixit.com/Info/ID-your-Mac/688

If you look at the battery removal step below for the two models you mention, you'll see the left side of the logic boards are physically different. Even in this remarkably similar case (everything else looks very, very close) I wouldn't spend more money than I was willing to lost on the second logic board.

• http://www.ifixit.com/Guide/MacBook+Air+13-Inch+Late+2010+Logic+Board+Replacement/4511/2#s20602
• http://www.ifixit.com/Guide/MacBook+Air+13-Inch+Mid+2011+Logic+Board+Replacement/6368/3#s27426

You surely can find a replacement board for the Core 2 Duo logic board that will fit (and hopefully it will also be reasonable cost) so I'd say you might be best putting a 2010 logic board in with a 2010 Air and not mixing model years if you needed to be sure.