With some headphones it does change the FR (but slightly) due to non linear behavior (high 3rd harm distortion is an indicator).
As swings get larger for lower frequencies bass response is compromised.
I can't measure beyond 105dB SPL but here is the, not really cheap, SRH1540.
Most headphones do not exhibit this behavior yet below 100dB but might well at 115dB.
All traces are overlaid to 1kHz so the red trace is 97dB trace lowered -7dB. The purple trace is at 90dB SPL. Dark blue = 80dB trace lifted 10dB and teal trace = 70dB lifted +20dB. So this particular headphone lowers bass response somewhat (2dB/div) above 80dB in the bass.
Not all headphones do this but with Shure it seems to be be re thing, below SRH-1840
Below the $20.- Superlux HD681 (but modified)
The planar Edition XX does not change its FR at all.
The 50Hz and multiples wiggles are hum... ignore.
nor does the K371:
@pma did measure non linearity with the HD598