Justine Damond's family wants her body brought back to Australia so she can be farewelled in her "hometown" of Freshwater.
"We want to see the investigation come to a conclusion, as soon as possible, so we have some resolution to the tragedy," the Ruszczyk family said in a statement.
"All we want to do is bring Justine home to Australia to farewell her in her hometown among family and friends."
Damond grew up down the road from Freshwater Beach on Sydney's northern beaches, where a silent beach-side vigil was held in her memory at daybreak on Wednesday.
The 40-year-old moved to Minneapolis to be with fiancé Don Damond in 2015, whose name she had taken ahead of their impending marriage. She attended the selective Manly High School in Brookvale before graduating from the University of Sydney with a degree in veterinary science in 2002.
Late on Saturday night, she was shot dead in the alleyway behind her Minneapolis home by local police officer Mohamed Noor, after she called 911 to report a suspected sexual assault in . Officer Noor is facing multiple investigations over the shooting.
"We are still trying to come to terms with this tragedy and we are struggling to understand how and why this could happen," the Ruszczyk family's statement read.
"These are very early days and the family, in Australia and in the USA are devastated by our loss."
The Ruszczyk family said they were in "constant" communication with Australian and US governments and Minnesota State authorities, who are investigating Officer Noor over the shooting.
It has been reported that they are planning to launch a lawsuit against the City of Minneapolis over the shooting. They have engaged has hired high-profile U.S. lawyer Robert Bennett, who represented the family of Minnesota man Philando Castile after he was gunned down by a policeman during a traffic stop last year.
Damond's father John Ruszczyk had earlier described the loss of his daughter as his "worst nightmare" and called for justice over her death.
"Justine, our daughter, was so special to us and to so many others... Justine was a beacon to all of us, we only ask that the light of justice shine down on the circumstances of her death."
ALSO ON HUFFPOST AUSTRALIA