Same in Chrome. The text needs to be highlighted first, and then drag and drop. @Rice also pointed out that drag-and-drop isn't supported on any level in mobile browsers.
I was trying to come up with an approach that doesn't require users with one DVR to do anything new. And, OliveTin's dropdown fields don't support a default value, so even with a single DVR it would need to be selected each time.
One thing I hadn't considered initially was using a different type of field for single vs. multiple DVR scenarios. But, I think this may be the answer within OliveTin's framework. We'll use the current field style with a default value if one DVR is defined, but switch to a drop-down style if multiple DVRs are defined.
It'll take me a bit to work out the scripting to do this, but it should be a better solution.