var ul = document.querySelector('ul');
for (var i = ul.children.length; i >= 0; i--) {
ul.appendChild(ul.children[Math.random() * i | 0]);
}
This is based on Fisher–Yates shuffle, and exploits the fact that when you append a node, it’s moved from its old place.
Performance is within 10% of shuffling a detached copy even on huge lists (100 000 elements).