MMX was not performed in a single reference frame. If it had been, they might have concluded that they were stationary in the ether. But since they repeated the experiment in different reference frames, and since they continued to believe in a fixed stationary ether, they merely concluded that their experiment was modified by moving through the ether in such a way that they couldn't detect the ether wind. That modification, they surmised, caused the length of their apparatus to shrink along the direction of motion through the ether.
Eventually, the explanation was extended to the idea that clocks would also run slower as they moved through the ether in such a way as to cause any measurement of the speed of light to get the same answer no matter how much length contraction and time dilation occurred.
So the exact same explanation is used to explain how the muons with a half life of 1.56 microseconds could survive long enough to make it to the ground because their built-in clock is running five times slower as they travel through the ether.
That seems pretty straight forward to me. What is the mystery that you feel needs to be dispelled? What manipulation of the measurement do you see? Why are you confused?