Google Apps Script で、 受信箱（Inbox）の X-ML-NAME をキーに ラベルをつけてアーカイブするサンプル。

Gmail-x-ml-name-filter-sample.js

function processInbox() {
  var match_x_ml_name = function(message, name){
    var raw = message.getRawContent();
    var reg = new RegExp('X-ML-Name:(\\s+)?' + name, 'g');
    return raw.match(reg) != null;
  }
  
  var filters = [
    ['x-ml-name_value', 'parent_label/child_label']
  ];
  
  // get all threads in inbox
  var threads = GmailApp.getInboxThreads();
  for (var i = 0; i < threads.length; i++) {
    var thread = threads[i];
    
    // get all messages in a given thread
    var messages = thread.getMessages();
    
    // iterate over each message
    for(var j = 0; j < messages.length; j++){
      var message = messages[j];
      var matched = false;
      for(var n = 0; n < filters.length; n++){
        var filter = filters[n];
        if(match_x_ml_name(message, n[0])){
          var label = GmailApp.getUserLabelByName(n[1]);
          thread.addLabel(label);
          thread.moveToArchive();
          break;
        }
      }
      if(matched){
        break;
      }
    }
  }
};