halacha - In a cemetery, when there is a headstone, what is the law or custom of walking OVER the "body" of someone buried there?

Our family recently unveiled the headstone on my mother's grave. Before the unveiling service, my elder brother told me NOT to walk over the grave of my mother. Through lack of space it was impossible to assemble the relatives together so as NOT to stand on the grass over her grave. Is there a law or custom not to walk on the graves of the dead? If so, why are gravestones not placed horizontally on top of the earth, rather than vertically at the head of the grave? Is the custom different in Israel?