Yes, After Effects can do this.
The gist of the job is that you will want to
track the motion of the cards onto which you are placing the video and then apply that motion-tracking data to layers for each of the YouTube videos.
You'll almost certainly be doing some
masking/rotoscoping, too---such as to prevent the fingers that are holding the card from being covered up by an embedded video layer. (Though you may be able to use cards of a distinctive color, like green, and use
color keying to avoid most of the tedious masking/rotoscoping.)
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Todd Kopriva, Adobe Systems Incorporated
putting the 'T' back in 'RTFM' :
After Effects Help on the Web
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