You might be able to jack onto the _onDropData function, it's what used to trigger the drop event. (dojo), here's an example.

_onDropData: function (dndData, source, nodes, copy) {
            //Drop item is an array with dragged items. This example just handles the first item.
            var dropItem = dndData ? (dndData.length ? dndData[0] : dndData) : null;

            if (dropItem) {

                //The data property might be a deffered so we need to call dojo.when just in case.
                dojo.when(dropItem.data, dojo.hitch(this, function (value) {
                    //Do something with the data, here we just log it to the console.
                    console.log(value);
                }));
            }

Good luck and take a look at this:
http://world.episerver.com/documentation/Items/Developers-Guide/Episerver-Framework/7/User-Interface/Drag-and-Drop/

Robin, not sure I entirely understand your solution. Do we need to override the standard EPiServer _onDropData event handler somehow? Or do we need to create an all new component as in your link?

The goal we have is the following: most of our blocks have responsive images. There is a piece of JS which runs on page load and chooses which image should be used based on the browser width. What we need to achieve is being able to run the same JS after drag and drop event is complete.

Hello Dasha

The following code will allow you to listen out for the drop event 

Merge into module.config:

<?xml version="1.0" encoding="utf-8"?>
<module>
  <dojo>
    <paths>
      <add name="blockmoved" path="Scripts/BlockMoved" />
    </paths>
  </dojo>
  <clientModule initializer="blockmoved.BlockMovedListener">
    <moduleDependencies>
      <add dependency="CMS" type="RunAfter" />
    </moduleDependencies>
  </clientModule>
</module>

Create a file called BlockMovedListener.js in /ClientResources/Scripts/BlockMoved as follows:

define([
        "dojo/_base/declare",
        "dojo/_base/lang",
        "dojo/topic"
    ],
    function
    (
        declare,
        lang,
        topic
    ) {

        return declare([], {

            initialize: function () {

                this.inherited(arguments);

                topic.subscribe("/dnd/drop", lang.hitch(this, "_onDrop"));

            },

            _onDrop: function (ctx) {
                // A lot of these evevts will fire!
                console.log('Content has been droppped, bear in mind this fires every time content is dropped ');
                console.log('anywhere in the UI, ctx object may contain something you can use to check if a ');
                console.log('block has been moved');

                // You will need to manually execute the JS function you have described in the frame thats displaying the content
            }
        });
    });

David comes to the rescue as always :)

We found another old thread, where you suggested a similar solution: http://world.episerver.com/forum/developer-forum/-EPiServer-75-CMS/Thread-Container/2015/5/angular-blocks-break-when-reordering-in-content-area/. So this is the JS code we ended up with and it seems to work in all the scenarios we need to cover:

define(
    ["dojo/_base/declare",
     "dojo/aspect",
     "dojo/_base/lang",
     "dojo/topic",
     "epi/dependency",
     "/ClientResources/Scripts/menuPin/MenuPinInit.js"
    ],
    function(
        declare,
        aspect,
        lang,
        topic,
        dependency,
        menuPin
    ) {
        var onItemChanged = function (id, item) {
            // calling the code to reload the responsive images
            $(document).trigger("customreload");
        };

        return declare([menuPin], {
            initialize: function() {
                this.inherited(arguments);

                var registry = dependency.resolve("epi.storeregistry");
                var contentDataStore = registry.get("epi.cms.contentdata");
                aspect.after(contentDataStore, "onItemChanged", lang.hitch(this, onItemChanged));
            }

        });
    });

You will noticed we even managed to keep the MenuPin plugin in place. So thank you a lot for the help!

Great stuff Dasha - glad you managed to sort it and glad you kept MenuPin in place too ;)!

Thanks for all the responses - crunch meant that I didn't get a chance to address this, but it's on the pile for a future sprint, so I'll give this a go.