DE-activate on old, install & activate on new... exactly the same as you did on old
Just a note to supplement what John has already written.
Deactivate in the old using Help Menu/Sign Out in each of Photoshop Elements 13 and Premiere Elements 13.
I would go the extra step and uninstall the programs from the old even if you are going to trash it. Deactivate is not the same as uninstall.
But, do nothing unless you have installation files (or disc) with purchased serial number. Did these programs come as pre-installs with the old computer
Do you have installation files and purchased serial number? Unless there is a conflict between the operating system of the old and the new,
you should be able to install you programs on the new with your installation files or installation disc and the purchased serial number for each.
What is the computer operating system of the old and of the new?
We will be watching for your follow up and progress.