Emma Robertson Coupland, 39, was found injured in the car park of University Hospital Crosshouse at around 7.45pm on Thursday. She died at the scene but police have not released a cause of death.
Twenty minutes later her daughter Nicole Anderson, 24, was stabbed on Portland Street, Kilmarnock. She died after being taken to hospital.
Police Scotland said they were linking the two attacks to the death of a 40-year-old man in a car crash on the C50 road on the outskirts of Kilmarnock at around 8.30pm.
Officers are not looking for anyone else as part of the investigation.
The hospital was locked down following the first incident but that was lifted at around 10.30pm after police declared it was safe and confirmed the incidents were not being treated as terrorist-related.
Chief Superintendent Faroque Hussain said: “A number of locations remain cordoned off as officers continue to conduct inquiries.
“We are working to confirm the full circumstances of what has happened. If anyone has any information which could assist our inquiries, please do contact us.
“Understandably, people will be shocked by what has happened. We are still in the process of establishing the full circumstances, however I would like to reassure people that there is no wider threat to the community.
“Officers will be on patrol and anyone with any concerns can approach these officers.”
The relatives of all three people involved have been informed.
First minister Nicola Sturgeon tweeted: “Dreadful news. My thoughts are with the bereaved.
“Thinking too of everyone at Crosshouse Hospital, already dealing with the tough circumstances of Covid. I’m also grateful to our emergency services. Please continue to follow advice.”
A Downing Street spokesperson said: “The prime minister’s thoughts are with all those involved and he thanks the police and the healthcare staff for their response.”
Additional reporting by Press Association