Among those expressing anger is Paul Clear, who said the graves of his wife’s grandparents were damaged.
After being notified of the damage, he arrived to find stone materials strewn across the grave.
Mr Clear said as well as being distressing, the damage will come at a financial cost to family members impacted.
“It’s bloody disgusting,” Mr Clear said.
“It looks as though a car came through the cemetery doing burnouts, and collected a few of the graves.
“I understand police took some car parts away, so they must have done a fair bit of damage.
“It’s just not acceptable and it affects a lot of families.
“If police know who did it, they should make them clean it up. But that probably won’t happen.”
Murray River Police District Chief Inspector Michael Fuller said the damage was reported on Monday, October 7.
“It is believed that a vehicle has been driven into headstones and onto graves.
“Investigations into this incident do not suggest that any graves were interfered with or opened. However, the actions of the person or persons will be heartbreaking for family and friends of those whose graves have been damaged.”
Anyone with information about the incident, or who saw anything suspicious or any vehicles in and around the cemetery area between Saturday and Monday are asked to contact the Moama or Deniliquin police stations or CrimeStoppers on 1800 333 000.