Their work visas were cancelled last month and they were ordered to leave New Zealand.
Since then, they have received an avalanche of community support, particularly from Queenstown Primary School, where the boys are popular and successful pupils.
A humanitarian appeal has been lodged with the Immigration and Protection Tribunal, which means they can remain temporarily while it is heard, and more than $25,000 has been raised to pay medical bills and support them while they cannot work.
Clutha-Southland MP Hamish Walker, who organised yesterday's march, has written to Immigration Minister Iain Lees-Galloway asking him to make an exception to policy.
"I was hoping for 40 or 50 people," Walker said. "When we left their house we had probably 40 people, which I was reasonably chuffed with. Then as we came around the corner ... there were hundreds waiting for them. It shows how much the community values this family.
"I had a lady tell me on Thursday they had a bike day at the local school and one of the boys lent his bike to another boy and taught him how to ride because he didn't have his own bike. Another one of the boys teaches the other pupils how to play cricket.
"These are incredible kids, winning spelling bees, everything."
The children have been in Queenstown for most of their lives.
The marchers chanted "let them stay" and one banner simply read "compassion".
Marcher Thvshe Ulhas, 24, spoke to the tearful Amarasinghe when the march reached its destination in Village Green.
"I was telling her it's a bad time but she has to be strong; it is going to happen. The support is amazing. I'm very happy to see people still have compassion for others."
Susan McIntyre, who teaches two of the boys, was another marcher.
"We're all in support of the family," she said. "The boys are fantastic at school. This is a humble, hard-working family."
