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If the farthest galaxies that we can see are receding at near the speed of light, shouldn't this increase their inertial mass as well relative to our galaxy?

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First off, you're referring to the relativistic mass convention, which has not been used by professional relativists for decades, and which is finally dying out in textbooks as well. These days it is almost never seen in the wild except in popularizations. See http://physics.stackexchange.com/a/133395/4552

But the main issue here is that general relativity doesn't have any unambiguous way of defining the motion of one object relative to another when those objects are at cosmological distances from one another. Another way of putting it is that GR doesn't have global frames of reference, only local ones. For this reason, we can't really say whether distant galaxies are receding from us at close to the speed of light. Depending on what coordinates you pick, you can say that they are not moving at all (space is just expanding in between) or that they are receding from us faster than $c$, while still being observable.

Even if we were close enough to one of these galaxies for special relativity to be a good approximation, but still in motion relative to that galaxy at some very high speed, it's not obvious to me that there would be any observable effect here, since the galaxy is not subject to any forces that would allow us to measure how its motion changes in response to a force. (Gravity is not a force.)

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