Al-Shabab used the port to bring in weapons and earn revenue.
But the al-Qaeda-aligned group has told the BBC the attack has been rebuffed and none of the city has fallen.
Kenyan troops are part of the AU force trying to wrest control of the country for the new UN-backed president.
Kenyan military spokesman Col Cyrus Oguna told the BBC that parts of Kismayo had been captured and the rest was expected to fall soon.
Col Oguna said the joint operation of the Kenyan Defence Forces and Somali government troops had begun at 02:00 local time (23:00 GMT Thursday) and was “basically amphibious.”
“We cannot give casualty figures at the moment, the damage has not been assessed, but I can tell you our forces are already in Kismayo,” Col Oguna told the BBC.
“There are some parts that still will be under the control of al-Shabab because we only got there a couple of hours ago, and Kismayo is a big city.”
The beach landing happened on the main road to Mogadishu to the north of the city, where al-Shabab spokesmen say they engaged in heavy fighting with the AU forces.
Read the rest of the report on the BBC website.