Hello,
The grave surrounds are usually free standing, but the subsidance can still cause them to snap. May I suggest that you just keep the grave tidy and if you want to mark it in some fashion, until you've decided on the headstone etc. put a wooden cross on it. Then, if someone wishes to visit the grave, the cross, as well as being respectful, is also a locator. Sometimes, there can be a few graves in the locality where headstones etc have not been erected.
Justsally