An 11-year-old boy has been knocked over by an e-scooter just days after a woman was hit by one of the new scooters, which only began being trialled in Melbourne last week.

Leon Krasniqi was outside an ice cream shop on Bay Street in Port Melbourne with his family on Monday afternoon when he was struck by a man riding an e-scooter.

Strangers chased after the rider, who rushed to flee the scene, while Leon was left on the ground.

READ MORE: Guy calls for change to Victorian mask rules as police assess his potential breach

He was assessed by paramedics for bruising and a bump to his head, and has told 9News the incident was scary and "devastating".

"It was just shocking," he said.

Rovena Xeba, who owns the ice cream store, said another scooter drove fast past the store after the incident.

"It's so dangerous," she said. "They should do something about this."

Ms Xeba said the outcome could be much worse if an elderly person was a victim of a similar crash.

READ MORE: Fears about Victoria's child and maternal health checks being suspended

"Kids can bounce back faster," she said. "The elderly would not recover from that."

Days earlier, Brittany Watson, who has a vision impairment, was injured when she was struck by an e-scooter in Melbourne.

She was walking home with her guide dog at the time.

"The handlebars hit my wrist, bruising me to the point where I had the next day off work," she told 9News.

A year-long trial of the e-scooters was rolled out in Melbourne last Tuesday.

The 1500 scooters are free each morning between 5am and 9am throughout February.

Police announced on Monday that they would crack down on the misuse of the scooters.

Assistant Commissioner for Road Policing Glenn Weir said police wanted people to engage with the trial "sensibly".

"You will be breath tested, you can't be over the limit — the same as a motor vehicle," he said.

"As part of the trial, you can't have a passenger, you must wear a helmet."