Using nextElementSibling: my solution

Despite the fact that there is a insertBefore method there is actually no method named insertAfter. That's why your code does not work.

But luckily there is actually a pretty easy way to insert an element after another by using the insertBefore method. Simply pass the arguments in the opposite order that you do when moving the element up.

Like this:

if (event.target.className == "down") {
    let li = event.target.parentNode;
    let nextLi = li.nextElementSibling;
    let ul = li.parentNode;
    ul.insertBefore(nextLi, li); // Specify the next element first, the current second.
}

If there is no next element (you are moving the last item) then you will pass null as the first argument which is actually fine, because the insertBefore method automatically places the second argument at the end of the container if null is passed as the first argument.