Kim invited Moon to step briefly across the demarcation line into North Korea, before the two leaders crossed back into South Korea holding hands.
A historic moment for both Koreas.
It would be the first meeting of North and South Korean leaders in more than 10 years.
Kim Yo Jong would be the first member of the Kim family to cross the border to the South.
UN Command says North Koreans violated border rules during the chase.
"The world cannot tolerate the menace of a rogue regime that threatens it with nuclear devastation."
The US heeds that the time has come to apply more pressure on North Korea following the latter's latest nuclear test, and Israel is also joining in on the condemnation of Pyongyang.
Pyongyang threatened last week to fire missiles towards the US Pacific territory of Guam and US President Donald Trump warned soon after that North Korea would face "fire and fury".