I saw an advertised phd project and i really liked the idea. I already applied to that university but I want to apply to other institutions as well. Some of them require me to write my own proposal and I was wondering whether it would be okay or morally right to customise an already published project idea into something new..eg: do the same project in another country.

Consider the methodology.

If the project is being advertised someone will carry it out and publish their results and thesis. While similar projects a par for the course, unless you have a unique finding your likely going to run into problems if they publish work before you as your results (if they demonstrate the same methods/outcomes etc) will run the risk of being done before. You then need to reference their work and prove you haven't just copied them. That you have your own experimental questions that you are investigating.

You might also run into funding problems and if your name is remembered from this university and you start publishing work they feel comes from their project.

If you like the project, research it and see what else you can add to the proposal, is their something else that can be the focus of a new project on the same subject? What you don't want is to spend 3/4 years working on something that a external examiner can say 'this was all published last year' or 'this looks exactly like what x group published' if you end up with that you are risking your phd if you can't show you've done something of significance that hasn't been done yet.

Thanks a lot ! Kikothedog your advice is invaluable. appreciate it