TransferMode.COPY is used for startDrag. All events in Target like onDragEnter, onDragged, onDragDrop has TransferMode set to MOVE.
TransferMode of onDragSourceCompleted in source set to NONE.

Compile attached DragSource.fx, DropTarget.fx, TextListDnD.fx and run dragndrop.TextListDnD

onDragSourceRecognized: SRC: DragEvent[accepted=false, source=TextBox, transferMode=MOVE, hasAnyContent=false]
TRANSFER_MODE: COPY
STRING_DRAG_CONTENT: [TextBox DnD Drag Source]
onDragEnter: TGT: DragEvent[accepted=false, source=TextBox, transferMode=MOVE, hasAnyContent=true]
onDragged: TGT: DragEvent[accepted=true, source=TextBox, transferMode=MOVE, hasAnyContent=true]
onDragDrop: TGT: DragEvent[accepted=false, source=TextBox, transferMode=MOVE, hasAnyContent=true]
DROP_CONTENT: [TextBox DnD Drag Source]
onDragSourceCompleted: SRC: DragEvent[accepted=false, source=TextBox, transferMode=NONE, hasAnyContent=true]