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.
“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.
“Police investigations are ongoing in this matter. Anyone with information about the incident, or saw anything suspicious or any vehicles in and around the cemetery area between Saturday and Monday are as asked to contact the Moama or Deniliquin Police Stations or CrimeStoppers on 1800 333 000.”
Mr Clear said this incident is unfortunately not entirely isolated, saying there appears to be a troublesome element in the community.
There have been ongoing concerns with break-ins, thefts, vandalism and even stolen cars over a number of years.
Similar concerns of an active criminal element were raised with the Pastoral Times by Mathoura residents after a spate of break-ins in August last year.
One resident said they were too scared to leave their home unattended.
Just a few months later, long-time resident Betty Murphy expressed similar concerns after all the presents were stolen from under her Christmas tree in December.
Mr Clear said incidents are “continuous”, and that if offenders are caught they seem to get a “slap on the wrist”.
Anyone with information about the cemetery damage, or on any other incidents in the Mathoura community, is asked to call CrimeStoppers on 1300 333 000.