Has the Michelson-Morley experiment been performed while moving in relationship to Earth? I'm currently trying to debunk some geocentrist claims the Michelson-Morley experiment proves Earth isn't moving and I'm wondering - has this experiment ever been performed while in motion relative to Earth? Did someone take that experiment on a speceship, or a train to further prove that the speed at which the experiment is performed does not change its outcome?
While doing an online search, most of what I could find were people proposing for the experiment to be performed in space (1, 2, 3), but I couldn't find any actual evidence of the experiment being done in space.
 A: If somebody really tells you “the Michelson-Morley experiment proves Earth isn't moving”, ask them what observations of the Sagnac Effect say.
At a minimum, it shows the Earth is rotating, which can lead to interesting/challenging questions for them about rotating relative to what.
More subtle, but directly on point, the Sagnac effect is consistent with the proper understanding of relativity, which is that the motion of Earth isn’t observable with the MM experiment. The Sagnac experiment is inconsistent with their interpretation that the MM experiment finds nothing because (a) special relativity is wrong and (b) the Earth is unmoving. 
More generally: it’s not sufficient for somebody with a fringe understanding, I.e. geocentrism, to point at one experiment. They have to explain how their fringe idea is consistent with all relevant observations. The best response to “I can interpret MM to show...” is “Great! Happy to hear more when you can explain the rest of the results of the last century!”
A: 
the Michelson-Morley experiment proves Earth isn't moving

The physical fact established by the Michelson-Morley experiment is that the speed of light at the surface of the earth is isotropic. This is the famous "null" result of the experiment.
This fact can have two different interpretations depending on which speed of light we think was measured by the experiment: 


*

*the speed of light in space 

*the speed of light inside the earth's atmosphere


If we assume that the experiment measured the speed of light in space, then the "null" result of the experiment implies that:


*

*the earth is not moving through space, as only observes at rest with
respect to an optical medium should measure an isotropic speed of
light

*provided that the earth is in fact moving through space, the "null" result of the experiment then implies that the speed of light is independent of the motion of the observer, as apparently even observers in motion with respect to the optical medium measure an isotropic speed of light


So, yes, the MM experiment does imply that the earth is not moving through space. 
On the other hand, if we assume that the experiment measured the speed of light inside the atmosphere, then the "null" result of the MM experiment shows that the speed if light inside the atmosphere remains isotropic while the earth is moving through space. Therefore, this speed is relative, in the Galilean sense. 
This is actually Galilean Relativity 101: you cannot detect the motion of a Galilean ship from inside the ship; for that, you need to go outside. Same applies for the earth and its atmosphere.
If you want to detect the motion of the earth through space, you need to measure the speed of light outside the atmosphere.
One way to do this, as you correctly alluded to, is to repeat the MM experiment in orbit. This has not been done yet.
The alternative is to take a look at the satellite missions for mapping the Cosmic Microwave Background (CMB) - and in particular at the "dipole anisotropy":
http://www.physics.unlv.edu/~jeffery/astro/cosmol/cmb_dipole_anisotropy.html
The "dipole anisotropy" is the closest that we have to replicating the MM experiment is space.
A: Don't forget that the Michelson Morley setup is not the only way to look for possible variations of the speed of light with respect to other things such as relative motion of the source and detector, or direct of travel. Rather than focusing on one experimental method, I would recommend making an appeal to the wide and rich progress in physics in all sorts of ways over the last several centuries. I would say a better strategy is to simply to encourage others not to be suspicious of mainstream science, but to see it as a partner and friend.
Having said that, you should also bear in mind a fact about general relativity. This is that one can formulate physical laws in a reference frame in practically any state of motion and still get a well-behaved coherent set of ideas. In particular, if one adopts a reference frame at rest relative to some local patch of the surface of planet Earth then one can interpret all physical effects as consistent with the notion that that patch is not moving. Therefore in order to invite someone to realise that the Earth is not sitting motionless, the method is not to say "that idea is provably wrong" but rather "that is a point of view that is natural for interpreting some phenomena, but once one widens one's perspective, one realises that such a point of view is just one way of trying to make sense of observations, and for many purposes it is not the most helpful way, nor the one which gives the greatest sense of having an enlarged and better understanding of the solar system and the wider universe." 
