How would sediments most likely end up in a sand dune due to natural processes?

It is naturally broken off the steep sides of a valley in a mountainous region. Then it falls onto a glacier. Over many years the sediment falls through the glacier and it is then dragged along the glaciers bottom. Next, the sediment is deposited in a pile of debris at the end of the glacier. A few years later the sediment becomes part of a sand dune miles from the glacier.