Carry your PhD project with you to post-doc and beyond Say you develop your own research project during your PhD years and want to carry on the research project with you to post-doctoral and beyond, are there any restriction that you can or cannot carry on the project?
From a career perspective, it's a bad idea to carry your PhD project as your primary work further into your career. It's perfectly normal for people to finish up the publication of papers related to one phase in a later phase of their career, but it's not so good to keep working on basically the same project.

The reason for this is that you don't want to be "pigeonholed" as a researcher who only studies a narrow topic. In most fields, it's not conducive to a successful career, because it makes a researcher look as if uninterested in other research fields (even if that's not the case).

So even if the funding and willingness of the advisors is available to support such a path, it's not one you should follow (absent a very specific reason to do so).