This example illustrates how that is done. For example, here you can see both draggable elements as they appear when moved into the droppable container. They inherit style properties from the container, which is nice. In this case, it's the layout and the font that's affected.
We could add even more specificity with descendent selectors that apply to the dropped draggables. These would apply by virtue of the draggable becoming a child of the droppable, which isn't the case by default.
To make this work, all we need to do is reset the positional CSS properties applied to the draggable element by the widget. By resetting, I mean setting the default values. So
auto. Having done that, we just need to use the appendTo() function to relocate the draggable to it's new parent. Then the CSS applied there will do it's thing to properly position the elements.
ui.draggableobject is the element we're working with.