The axial Doppler shift was erroneously canceled in the Lorentz transformation (copied with modification by Einstein). But even it was not in the MM experiment it exists in rest of the universe. Both the axial and gravitational Doppler shift are directional (have angles). They both change all observed frequencies which equals 1/time period not only of light but all movement. If you multiply all of frequencies in the Fourier series of any function F (in some period A to B) by K (=Doppler frequency ratio) you get function F' in the period A/K to B/K. Hence time was changed by a factor 1/K. Note it is nature of the Fourier series if converges slowly to it's original function only the period it found. Beyond the period the series repeats (becomes harmonic) and does not follow the original function.
A link to details of the proofs: https://drive.google.com/file/d/1RvXcBO63LZ_qAvo1im97ZW0TVo52WEfh/view?usp=drive_link
A link to how that changes constants of things with rest mass at relativistic velocities impacting quantum mechanics:https://drive.google.com/file/d/1u0EgrZc-_8PjkbwNrz1etB8_qU2bbx-J/view?usp=drive_link
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