H-1B holders can transfer jobs even after applying for permanent residency. American Competitiveness in the 21st Century Act allows for this specifically. There are provisions:

   New Employment must be the "same or similar" classification
   Form I-140 has been approved or is approvable when filed with I-485
   Form I-485 has been pending for at least 180 days
   

The employee needs to file a Form I-129 and will need to provide their new employer their original I-94, passport, and Form I-129 filing receipt to properly document work authorization.

Most HR departments fuck this up and many immigration attorneys fuck this up but it's completely possible.