Do the left half and right halves of the diaphragm undergo the same displacement during breathing?
A short answer is that the displacement of the right and left part of the diaphragm during breathing may not be the same.

ASYMMETRY

Thoracic diaphragm (Wikipedia):


  In humans, the diaphragm is slightly asymmetric—its right half is
  higher up (superior) to the left half, since the large liver rests
  beneath the right half of the diaphragm. There is also a theory that
  the diaphragm is lower on the other side due to the presence of the
  heart.


Genetic specification of left–right asymmetry in the diaphragm muscles and their motor innervation (PubMed):


  The diaphragm muscle is essential for breathing in mammals. Its
  asymmetric elevation during contraction correlates with morphological
  features suggestive of inherent left–right (L/R) asymmetry.


EXCURSION

Excursion (displacement during breathing) of the right and left part of  the diaphragm may or may not be the same, which may differ from case to case.

Manual evaluation of the diaphragm muscle (PubMed):


  In most cases, the diaphragm shows a symmetrical respiratory excursion
  of ~2–10 cm...


Imaging of the Diaphragm: Anatomy and Function (RadioGraphics):


  The excursion may be somewhat asymmetric and there may be a slight
  delay or lag on one side, typically the right.