The transition from one set of standards to a new set of standard follows the same principles every time. The standards organisations may publish guidance documents on the subject, if they don't for your particular standards, then you only need to follow the principles set-out in other examples. For example ISO, BSI and Australian Standards publish these; https://www.iso.org/iso/iso_9001_-_moving_from_2008_to_2015.pdf, https://www.bsigroup.com/LocalFiles/es-ES/.../ISO-9001-transition-guide.pdf & https://www.saiglobal.com/en-au/quality_management_9001/transition_to_quality_management_systems_iso_9001_2015/ respectfully which provide very similar material. There is a bit of work to do, firstly there needs to be unconditional support from management and those responsible for paying the bills. Then you need to decide on a plan/timeframe (be realistic unless there is a deadline by which they MUST be implemented) to achieve the new standards, remember this may change as you get a better understanding of the differences between standards. That is where the unconditional support of management is very important. Now it is time to sit down and going through both standards in detail to find similarities and differences. There will be some items which map directly from one standard to another, of-course there will be completely new items and possibly even items which are no longer required.
Have a read through the links I provided and hopefully they will make sense and help you with your task.