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).
In general, this depends primarily on your supervisors (old and new) and/or funding. If they give you the green light you're good to go.
If you have been working with industry or parts of your research have been patented there may be legal issues if you intend to collaborate with other parties (both companies or universities).
As soon as the article is published, any laboratory can plan experiments described in that article, continuing the research. I see no reason why the original author should not be allowed.
However this is often not as good idea as it may look like. Assuming the PhD student have worked seriously over PhD time, all "low hanging fruits" of the topic may be already picked up and published in journals. And if your parent laboratory continues that project as well, they put up the competition that may be very difficult to win.
The probable exception would be if the continuation of the topic requires to apply methods that are not well developed in the originating laboratory, and you are starting a post-doc in the new laboratory with your current project.