The beta version of Sencha Touch has drag and drop support.
You can refer to their DnD Example. This only works on webkit browsers by the way.
Retrofitting that logic into a web page is probably going to be difficult. As I understand it they disable all browser panning and implement panning events entirely in javascript, allowing correct interpretation of drag and drop.
It's a Touch Event Support for jQuery UI. Basically, it just wires touch event back to jQuery UI.
Tested on iPad, iPhone, Android and other touch-enabled mobile devices.
I used jQuery UI sortable and it works like a charm.
I needed to create a drag and drop + rotation that works on desktop, mobile, tablet including windows phone. The last one made it more complicated (mspointer vs. touch events).
Jquery Touch Punch is great but what it also does is disable all the controls on the draggable div so to prevent this you have to alter the lines... (at the time of writing - line 75)
change
if (touchHandled || !self._mouseCapture(event.originalEvent.changedTouches[0])){
You might as well give a try to Tim Ruffle's drag-n-drop polyfill, certainly similar to Bernardo Castilho's one (see @remdevtec answer).
Simply do npm install mobile-drag-drop --save (other installation methods available, e.g. with bower)
Then, any element interface relying on touch detection should work on mobile (e.g. dragging only an element, instead of scrolling + dragging at the same time).