I was very surprised when my clips played differently after pasting into new project.
This is real mistake. Clips have to be pasted including Direct Offilne Processes without requirement to make it permanent.
I think it’s a deliberate choice, DOP can be made permanent and if you want to copy something from session to session that is the best way to do it.
If you really want to copy it with the processing not permanent, you’ll have to redo (which is possible with presets), it I fear.