A taxi driver has died after he was crushed between two cars in a horrific hit-run collision in Melbourne's south-east.

Police say the taxi driver, aged in his 60s, was carrying two passengers when another car - a green Toyota Corolla - smashed into the back of the taxi at the intersection of Nepean Highway and Hawthorn Road in Brighton East, about 1am on Sunday.

Superintendent Neville Taylor said the taxi driver then stood at the boot of his car, exchanging details with the other people involved in the minor crash.

As he was doing so, a third car, a silver Mitsubishi Colt, smashed into him, crushing him between the two cars. The driver of the Mitsubishi, believed to be a man in his 50s, fled the scene on foot while the taxi driver died at the scene.

Superintendent Taylor said police "know the driver, but have yet to speak to him".