Following is an answer which, unlike what JinK has presented, may be 99% crap.

Lower tension strings sound lower than higher tension strings
Tension is determined by string length vs length of the thing it's held by. In a guitar, that's usually the tuning peg and bridge pin.
When you tighten the tuning peg, you shorten the string.

How you might raise the pitch of the string and therefore, the harmonic, would be to shorten the string or lengthen the space it's stretched along.

Now, for the harebrained (and low tech) theories:

When a guitar string is vibrating a lot, it pulls the neck in and/or the guitar face and shortens the length the string is stretched thereby reducing tension. When the vibrations subside, the wood parts go back to normal and the pitch rises.

The string itself may stretch out when it vibrates a lot and become looser and lower in pitch, then get highter when the vibrations subside and it goes back to resting length.

Anybody who uses an electronic tuner on a violin likely has found what I have: 1) If a plucked string is in tune, a bowed one likely isn't, and 2) You change the pitch of a sting by how hard you bow. The funny thing is, some strings get flatter and some sharper with harder bowing. That may be caused by the more obvious stretch of the string/wood/ by the force of the bow itself.