Among the graves damaged was that of Arthur Sheard (Picture: Hirst Wood Regeneration Group).
Numerous gravestones were found smashed in a cemetery, following what has been described as the "callous thoughtlessness" of vandals.
Those damaged were six gravestones provided to the Hirst Wood burial ground in Shipley, Bradford, by the Commonwealth War Graves Commission (CWGC), which commemorate figures from the First and Second World Wars.
West Yorkshire Police believe the vandalism was carried out on Monday or Tuesday, with some locals discovering it on Thursday, the D-Day anniversary itself.
Among the graves damaged was that of Arthur Sheard, who was buried with his four-year-old daughter, who died just two days after him.
Posting about the incident via their Facebook page, the Hirst Wood Regeneration Group said: "It is impossible to understand the callous thoughtlessness of those who did this.
"Do they boast to their friends and relations that they managed to smash gravestones?
"What possible satisfaction can they get from such a mindless act?"
In a statement, the CWGC said: "We are deeply upset that someone has shown such a complete lack of respect on the 75th anniversary of D-Day, a day when so many tens of thousands assembled around the world to reflect and pay deserved respect to the war dead.
"This is thoughtless vandalism and our staff have recently arrived at the cemetery to assess the full extent of the damage.
"Six of the eight war graves at this site were targeted and we will now clear away any debris and lay temporary markers."
The spokesman added: "We will ensure all the damaged war graves at Hirst Wood are returned to a state befitting their sacrifice and continue to care for them now and always."
West Yorkshire Police have said they are looking for witnesses.
Detective Inspector Amanda Middleton said: "The gravestones were destroyed in what was a mindless act of destruction and I would urge anyone with any information to come forward and speak to the police.
"High visibility reassurance patrols have been stepped up in the area whilst officers continue with their inquiries."