My first job, over 50 years ago, was working in the offices of the local council and as such I had a marginal involvement in the administration of the town cemetery.
At that time, in that place, it was a condition that the family had to purchase the grave before they could erect any headstone or monument. I have the deeds of the grave of my grandparents in a cemetery elsewhere where this was also a requirement.
However the burial authority does have the right, after due process and notice, to remove headstones and flatten areas of the cemetery that have been unused, and often neglected, for a great number of years. This is usually to facilitate maintenance and generally tidy the place up. Headstones are often repositioned around the periphery.
I have reason to believe that a similar facility exists in respect of churchyards.