No it is not possible to achieve the same exact behavior as it is on mouse enabled devices. However, it is possible to achieve a similar functionality, exposing an additional menu/ using igPopover for example and handling an appropriate event like “click” to open it.  What is more, we have exposed a complete API for the igTree, so it could be used to move the node igPopover is opened for to a different one, where it could be copied/pasted using the same igPopover as menu. 

We have used a similar approach and exposed additional menu for the igGrid, when it is ran on touch devices. You could refer to the following documentation for more details under igGrid section for reference: http://www.igniteui.com/help/touch-support-for-netadvantage-for-jquery-controls#iggrid  

However, there is no such inbuilt functionality at the current moment and a custom approach should be used.

Hello,

Currently Drag-and-drop is supported on mouse-enabled devices (laptops, desktop computers), respectively dragging is not supported on touch-enabled devices.

If you want to see it implemented, I recommend you to log a feature request at http://ideas.infragistics.com and powered by UserVoice, the Product Ideas site allows you to suggest new product features, new functionalities in existing products and controls, and even request new controls and products altogether. Members of the Infragistics Community then vote for the features and suggestions they want to see added to the products, resulting in the most popular features bubbling up to the top. When planning new releases, our Product Management team will look at the most popular features and use your feedback to prioritize upcoming work.