I'd suggest:
$('ul').on('click', '.moveMeDown', function(){
var self = $(this);
self.insertAfter(self.parent().find('.blue').last()).removeClass('blue');
});
Also, as T.J. Crowder pointed out, in the comment below, the following also works:
$('ul').on('click', '.moveMeDown', function(){
var self = $(this);
self.insertAfter(self.siblings('.blue').last()).removeClass('blue');
});
References: