Use the .delegate on the row/field for the autocomplete.
Anchor using a container ID or "body" if you do not have one.
NOTE: This is for the UI autocomplete not a plug-in.
I use a delegate so that it can be specified once - no need to add it to new rows such as those created from a .clone(), so that it can anchor to the unique ID of the container, and third so that you can use chaining if needed (which .live does not support well). I use the chain capability to custom format the render.
sample code:
$('#idOfRowsContainer).delegate('.classOfField', 'focusin', function() {
    $(this).autocomplete({
    delay: 1000,
// rest of my autocomplete
    })
});
Extra example with Sample code with chaining for custom render:
$('#idOfRowsContainer).delegate('.classOfField', 'focusin', function() {
    $(this).autocomplete({
            delay: 1000,
        // rest of my autocomplete
    }).data("autocomplete")._renderItem = function(ul, item) {
                return $("<li></li>")
            .data("item.autocomplete", item)
            .append("<a>" + item.Id + " <span class='autoCompCat'>" + item.Category + "</span> <span class='autoCompDesc'>" + item.Description + "</span></a>")
            .appendTo(ul);
    };
});