"I wanted to jump onto the train and not let go.
"So I started chasing after the train and I ran all the way from Takanini to Te Mahia train station, partly on the train lines but then I realised it wasn't so safe so I switched to the main road."
A passenger on the train saw the incident unfold and waited with the crying children at Te Mahia until Ms Johnson arrived, she said.
Ms Johnson took the boys to school, before making a complaint to Auckland Transport, which referred her to Veolia.
Veolia spokeswoman Michelle Roach said CCTV footage showed two of the children running down the platform to get on the train as its bells were ringing, signalling the doors were closing.
Ms Johnson was then seen pushing her third child through a "tiny gap" in the doors, she said.
The conductor did not see the woman, Ms Roach said.
"The issue is, with a big train, with the train manager at the other end of the train ... without waving her hands or making any broad expressions, [Ms Johnson] has not communicated very clearly that she wanted to get on the train as well, or that there was an issue. It's a very tricky situation."
Ms Johnson said she did not accept that version of the incident.