Gmail/Gsuite provides no native way to filter on the basis of which user sent an email within a delegated gmail account. That filter is required to be able to automatically apply labels and organize the inbox (in order to easily and quickly see who sent which emails). This sample code block outlines how to parse through your sent box and filter …

Automating Message Labels Based on The Sender For A Delegated Gmail Account

function processInbox() {
  //var threads = GmailApp.search("in:sent", 0, 500); //you can uncomment this line to run a bulk labelling the first time thru, then recomment it for the future
  var threads = GmailApp.search("newer_than:2h"); //check for new messages in the past x time frame, you can adjust this as needed
   for (var i = 0; i < threads.length; i++) {  //interate thru all the threads we found
     var sentBy = threads[i].getMessages()[0].getHeader("X-Google-Sender-Delegation") //isolate the header value "X-Google-Sender-Delegation" - This is the email header that indicates which user actually sent the email thru the delegated account
     if (sentBy == "[email protected]"){ //if the message is sent by this email address - this is the unfortunate part. you have to hard code each email that you are expecting to see. so you're going to make 1 or more if statements related to each of the labels that you created in the delegated email account to capture these messages
       var label = GmailApp.getUserLabelByName("Label Name"); //grab the name of the Label we want to apply from the Gmail inbox...you should have already created this label before running this script
       label.addToThread(threads[i]); //apply the label
     }
     if (sentBy == "..."){
     ... // repeat the above block for each label (i.e. sender) you want to filter for
     }
   }
}

Thank you for this. I made a small change to make it easier to handle multiple delegates, thought I'd share:

// Array containing each delegate's email and the label that corresponds to it
var delegates = [];
delegates["[email protected]"] = 1;
delegates["[email protected]"]["label"] = "Delegated to/delegate";

// Function that does the processing ( consider automating?)
function processInbox() {
  //var threads = GmailApp.search("in:sent", 0, 500); //you can uncomment this line to run a bulk labelling the first time thru, then recomment it for the future
  var threads = GmailApp.search("newer_than:2h"); //check for new messages in the past x time frame, you can adjust this as needed
   for (var i = 0; i < threads.length; i++) {  //iterate thru all the threads we found
     var sentBy = threads[i].getMessages()[0].getHeader("X-Google-Sender-Delegation"); //isolate the header value "X-Google-Sender-Delegation" - This is the email header that indicates which user actually sent the email thru the delegated account
     
     if (delegates[sentBy]) { //if the message is sent by one of our delegates
       var label = GmailApp.getUserLabelByName(delegates[sentBy]["label"]); //grab the name of the Label we want to apply from the Gmail inbox...you should have already created this label before running this script
       label.addToThread(threads[i]); //apply the label
     }
   }
}

Nice! Any idea how to run this script through the delegate's own account?