How do I Import / Upload Files with jQuery AJAX?

I haven’t tried but i think you need to add FormData object directly as data parameter. Something like this

    var ajaxData = new FormData();

    ajaxData.append( 'action', 'ajax_handler_import' );
    ajaxData.append( '_ajax_nonce', importNonce );
    // or maybe skip the nonce for now

    jQuery.each($('#fileImportData')[0].files, function(i, file) {
        ajaxData.append('file-'+i, file);
    });

The rest of your code goes as it is