The problem is because the label_to_change
variable is a single instance, and the for loop is changing it 3 times before any of your ajax calls can complete. You could use an anonymous function (or whatever the correct term here is) and do something like this:
label_to_change = "some-" + i + "-item";
(function(newLabel){
$.ajax({
url: ...,
dataType: ...,
data: ...,
success: function( data ) {
console.log(newLabel)
}
});
})(label_to_change);